A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

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<p>The diagram shows the graph of $y=\mathrm{f}(x)$, which has a minimum point at $(-3,-3)$. The curve is such that both the $x$ and $y$-axis are asymptotes, and $y=-2x$ is an oblique asymptote.</p>
<ol class="abc-list"><li data-list-item-id="imported-item-block-2-item-0" data-list-item-path="imported-item-block-2-item-0"><div class="list-item-paragraph"><p>State the range of values of $x$ for which $\mathrm{f}$ is decreasing.</p></div></li><li data-list-item-id="imported-item-block-2-item-1" data-list-item-path="imported-item-block-2-item-1"><div class="list-item-paragraph"><p>Sketch the graph of $y=\mathrm{f}'(x)$.</p></div></li></ol>

The diagram shows the graph of $y=\mathrm{f}(x)$, which has a minimum point at $(-3,-3)$. The curve is such that both the $x$ and $y$-axis are asymptotes, and $y=-2x$ is an oblique asymptote.

Interpreting Gradients and Concavity

Sketching Derivative Graphs

Applications of Differentiation

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>Given that&nbsp;$(3,0)$ is a point of inflection of $y=\mathrm{f}\left( x \right)$. State the maximal range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is </p><ol class="abc-list"><li data-list-item-id="item-1782721008576-2k1azs74n" data-list-item-path="item-1782721008576-2k1azs74n"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1782721009392-41xoixgky" data-list-item-path="item-1782721009392-41xoixgky"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1782721009848-wlrux4odu" data-list-item-path="item-1782721009848-wlrux4odu"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1782721010480-idm7y5b1r" data-list-item-path="item-1782721010480-idm7y5b1r"><div class="list-item-paragraph">concave downwards.</div></li></ol></div></div></div>

Transformations of Graphs

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>State the maximal range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is </p><ol class="abc-list"><li data-list-item-id="item-1782721448758-9v2vlclnm" data-list-item-path="item-1782721448758-9v2vlclnm"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1782721449230-9q3crfsy1" data-list-item-path="item-1782721449230-9q3crfsy1"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1782721449750-4vvlkis5t" data-list-item-path="item-1782721449750-4vvlkis5t"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1782721450214-fmkc17s80" data-list-item-path="item-1782721450214-fmkc17s80"><div class="list-item-paragraph">concave downwards</div></li></ol></div></div></div>

<p></p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1771834571585-e61ojqrom1v.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE4MzQ1NzE1ODUtZTYxb2pxcm9tMXYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEwMSwiZXhwIjoxNzkyODM5NDAxfQ.svbZO5ad97knO_sOQLBXm3iZiawCNPzDLkqpG3eJh4s" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y=\mathrm{f}'(x)$.</p><ol class="abc-list"><li data-list-item-id="item-1782722076759-ajsrwfjnl" data-list-item-path="item-1782722076759-ajsrwfjnl"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y=\mathrm{f}(x)$. Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1782722077575-ubahx25n0" data-list-item-path="item-1782722077575-ubahx25n0"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y=\mathrm{f}(x)$ is,</div><ol class="i-list"><li data-list-item-id="subitem-1782722080063-mrocqlaob" data-list-item-path="item-1782722077575-ubahx25n0/subitem-1782722080063-mrocqlaob"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1782722080615-72ftoujks" data-list-item-path="item-1782722077575-ubahx25n0/subitem-1782722080615-72ftoujks"><div class="list-item-paragraph">decreasing.</div></li></ol></li></ol></div></div></div>

<p></p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1782721921357-z8kwh67ipxj.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI3MjE5MjEzNTctejhrd2g2N2lweGoucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEwMSwiZXhwIjoxNzkyODM5NDAxfQ.-uWERSP2wKykKrTJmNqhRbR7ktvhrfHScKOZr43jn9Y" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y=\mathrm{{f}'}\left( x \right).$</p><ol class="abc-list"><li data-list-item-id="item-1782721816459-dh3lt08na" data-list-item-path="item-1782721816459-dh3lt08na"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y=\mathrm{f}\left( x \right).$ Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1782721817539-pknrkfaxx" data-list-item-path="item-1782721817539-pknrkfaxx"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y=\mathrm{f}\left( x \right)$ is,</div><ol class="i-list"><li data-list-item-id="subitem-1782721831084-1nxhba7ch" data-list-item-path="item-1782721817539-pknrkfaxx/subitem-1782721831084-1nxhba7ch"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1782721831547-jgq9e8n15" data-list-item-path="item-1782721817539-pknrkfaxx/subitem-1782721831547-jgq9e8n15"><div class="list-item-paragraph">	increasing. </div></li></ol></li></ol></div></div></div>

Stationary Points of Inflection

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y=\mathrm{{f}'}\left( x \right).$</p><ol class="abc-list"><li data-list-item-id="item-1782722392059-3w61e5l81" data-list-item-path="item-1782722392059-3w61e5l81"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y=\mathrm{f}\left( x \right).$ Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1782722392924-80t3z2p82" data-list-item-path="item-1782722392924-80t3z2p82"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y=\mathrm{f}\left( x \right)$ is,</div><ol class="i-list"><li data-list-item-id="subitem-1782722394931-3xmwty7or" data-list-item-path="item-1782722392924-80t3z2p82/subitem-1782722394931-3xmwty7or"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1782722395075-lpmoz48d1" data-list-item-path="item-1782722392924-80t3z2p82/subitem-1782722395075-lpmoz48d1"><div class="list-item-paragraph">strictly decreasing. </div></li></ol></li></ol></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y=\text{{f}'}\left( x \right).$</p><ol class="abc-list"><li data-list-item-id="item-1782722258562-wsz8r0qr6" data-list-item-path="item-1782722258562-wsz8r0qr6"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y=\mathrm{f}\left( x \right).$ Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1782722259353-1djncg7ng" data-list-item-path="item-1782722259353-1djncg7ng"><div class="list-item-paragraph">State the maximal range of values of $x$such that the curve $y=\mathrm{f}\left( x \right)$ is,</div><ol class="i-list"><li data-list-item-id="subitem-1782722269178-hpq0kg2vm" data-list-item-path="item-1782722259353-1djncg7ng/subitem-1782722269178-hpq0kg2vm"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="subitem-1782722269562-ocnvsljxp" data-list-item-path="item-1782722259353-1djncg7ng/subitem-1782722269562-ocnvsljxp"><div class="list-item-paragraph">increasing. </div></li></ol></li></ol></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1782721203078-j492fx63cgs.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI3MjEyMDMwNzgtajQ5MmZ4NjNjZ3MucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEwMSwiZXhwIjoxNzkyODM5NDAxfQ.hOl0llfxaTRC1sf8GBUCCjP-xIY4MWBNdOuzHxIXOKM" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>State the maximal range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is </p><ol class="abc-list"><li data-list-item-id="item-1782721154762-r9vgfqafe" data-list-item-path="item-1782721154762-r9vgfqafe"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1782721155338-80xbhqtyw" data-list-item-path="item-1782721155338-80xbhqtyw"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1782721155866-gy1wq91f7" data-list-item-path="item-1782721155866-gy1wq91f7"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1782721156466-1lum0iwd3" data-list-item-path="item-1782721156466-1lum0iwd3"><div class="list-item-paragraph">concave downwards.</div></li></ol></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1771831679646-gzw18dhibk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzE4MzE2Nzk2NDYtZ3p3MThkaGliay5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTAxLCJleHAiOjE3OTI4Mzk0MDF9.l3OH_1oTjMN1XhySG705E1z9NkYRyFVQo_vNc8nP5wI" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>State the maximal range of values of $x$ for which the graph of $y=\mathrm{f}\left( x \right)$ is </p><ol class="abc-list"><li data-list-item-id="item-1782721008576-2k1azs74n" data-list-item-path="item-1782721008576-2k1azs74n"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1782721009392-41xoixgky" data-list-item-path="item-1782721009392-41xoixgky"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1782721009848-wlrux4odu" data-list-item-path="item-1782721009848-wlrux4odu"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1782721010480-idm7y5b1r" data-list-item-path="item-1782721010480-idm7y5b1r"><div class="list-item-paragraph">concave downwards.</div></li></ol></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="300" data-height="200" style="width:300px;height:200px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y=\mathrm{f}'(x)$.</p><ol class="abc-list"><li data-list-item-id="item-1782723992082-4d7drdhph" data-list-item-path="item-1782723992082-4d7drdhph"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y=\mathrm{f}(x)$. Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1782723992794-f877iijtx" data-list-item-path="item-1782723992794-f877iijtx"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y=\mathrm{f}(x)$ is,</div><ol class="i-list"><li data-list-item-id="subitem-1782723995082-fmvff8tlv" data-list-item-path="item-1782723992794-f877iijtx/subitem-1782723995082-fmvff8tlv"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1782723995386-v3877d72h" data-list-item-path="item-1782723992794-f877iijtx/subitem-1782723995386-v3877d72h"><div class="list-item-paragraph">strictly decreasing.</div></li></ol></li></ol></div></div></div>

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="250" data-height="250" style="width:250px;height:250px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y=\mathrm{f}'(x)$.</p><ol class="abc-list"><li data-list-item-id="item-1782722878082-7qorrsc01" data-list-item-path="item-1782722878082-7qorrsc01"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y=\mathrm{f}(x)$. Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1782722879434-1yvgzgzn8" data-list-item-path="item-1782722879434-1yvgzgzn8"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y=\mathrm{f}(x)$ is,</div><ol class="i-list"><li data-list-item-id="subitem-1782722881259-9vgcqnm5m" data-list-item-path="item-1782722879434-1yvgzgzn8/subitem-1782722881259-9vgcqnm5m"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1782722881729-dl5367023" data-list-item-path="item-1782722879434-1yvgzgzn8/subitem-1782722881729-dl5367023"><div class="list-item-paragraph">strictly increasing,</div></li><li data-list-item-id="subitem-1782722882010-onol01kz2" data-list-item-path="item-1782722879434-1yvgzgzn8/subitem-1782722882010-onol01kz2"><div class="list-item-paragraph">decreasing.</div></li></ol></li></ol></div></div></div>

<p>The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768702377706-0fn6fgzv11j7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njg3MDIzNzc3MDYtMGZuNmZnenYxMWo3LnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI4MzkxMDEsImV4cCI6MTc5MjgzOTQwMX0.eL7mFfDO19fDwMZ11KMg0i0CAMetbajV-rsASghBFtw" style="max-width: min(800px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">State the nature of all turning point(s) of the graph of $y=\mathrm{f}(x)$.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">State the range of values of $x$ where&nbsp;$\mathrm{f}$ is decreasing.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}(x)$ indicating clearly the equations of the asymptotes, coordinates of the turning point(s) and the intersections with the axes.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">On the copy of the graph of $y=|\mathrm{f}(x)|$ in the Printed Answer Booklet, sketch and label a line $y=kx+3k$, where&nbsp;$k$ is a constant. Hence state the range of values of $k$ for which there is no real solution to the equation $|\mathrm{f}(x)|=kx+3k$.<span class="inline-marks">[2]</span></div></li></ol>

The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below.

State the nature of all turning point(s) of the graph of $y=\mathrm{f}(x)$.

State the range of values of $x$ where $\mathrm{f}$ is decreasing.

Sketch the graph of $y=\mathrm{f}(x)$ indicating clearly the equations of the asymptotes, coordinates of the turning point(s) and the intersections with the axes.

On the copy of the graph of $y=|\mathrm{f}(x)|$ in the Printed Answer Booklet, sketch and label a line $y=kx+3k$, where $k$ is a constant. Hence state the range of values of $k$ for which there is no real solution to the equation $|\mathrm{f}(x)|=kx+3k$.

Parameter Conditions for Graphs and Equations

Increasing and Decreasing Functions

Graphs of Modulus Function

Stationary Points and Points of Inflection

2025 DHS P1 Q5 

Graphing Techniques

<p>The diagram below shows the curve $y = \mathrm{f}'(x)$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<div class="graph-block" data-graph-config="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" data-width="400" data-height="350" style="width:400px;height:350px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y = \mathrm{f}(x)$. Hence determine the nature of these stationary points.</div></li><li><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y = \mathrm{f}(x)$ is</div><ol class="i-list"><li><div class="list-item-paragraph">concave downwards,</div></li><li><div class="list-item-paragraph">strictly decreasing.</div></li></ol></li></ol>

The diagram below shows the curve $y = \mathrm{f}'(x)$.

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="400" data-height="350" style="width:400px;height:350px;max-width:100%;display:block;margin:0 auto;"></div></div><div class="question-column" style="flex: 1; align-self: flex-start;"><p>The diagram below shows the curve $y = \mathrm{f}'(x)$.</p><ol class="abc-list"><li data-list-item-id="item-1783334124942-vktabksi4" data-list-item-path="item-1783334124942-vktabksi4"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y = \mathrm{f}(x)$. Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1783334126037-dvve25yxw" data-list-item-path="item-1783334126037-dvve25yxw"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y = \mathrm{f}(x)$ is</div><ol class="i-list"><li data-list-item-id="subitem-1783334130022-wyfnbqzmg" data-list-item-path="item-1783334126037-dvve25yxw/subitem-1783334130022-wyfnbqzmg"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1783334130750-9jv93f7vh" data-list-item-path="item-1783334126037-dvve25yxw/subitem-1783334130750-9jv93f7vh"><div class="list-item-paragraph">increasing.</div></li></ol></li></ol></div></div></div>

<p>The diagram below shows the curve $y = \mathrm{f}'(x).$</p>
<div class="graph-block" data-graph-config="eyJheGlzIjp0cnVlLCJncmlkIjpmYWxzZSwid2lkdGgiOjQwMCwiaGVpZ2h0IjozNTAsImVsZW1lbnRzIjpbeyJpZCI6IjVmNDNhMzFkLTk2NzQtNGFlYi04OTdlLTNiZDYxOWIyM2M3NCIsImRhc2giOjIsInR5cGUiOiJzZWdtZW50IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCh4PTFcXCkiLCJwb2ludDEiOlsxLC00LjVdLCJwb2ludDIiOlsxLDVdLCJsYWJlbE9mZnNldCI6Wy03LC0yMzBdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiZTE0OTliNmYtNzdkOC00YTg0LWEwMDgtYjY3YjA3NWFkZWU2IiwiZGFzaCI6MiwidHlwZSI6InNlZ21lbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKHk9LXgtMVxcKSIsInBvaW50MSI6Wy01LDRdLCJwb2ludDIiOlszLjUsLTQuNV0sImxhYmVsT2Zmc2V0IjpbMTgwLC0yMDBdLCJzdHJva2VXaWR0aCI6MS41fSx7ImlkIjoiOWFhMjE4NDYtMmQwOS00OGU5LTg3ZDMtZGJlYjE0ZDIxZTNiIiwidHlwZSI6ImJlemllciIsImNvbG9yIjoiIzAwMDAwMCIsInBvaW50cyI6W1stMi42MiwxLjIxXSxbLTAuNzQsLTAuMzhdLFswLjc2LC0yLjYzXSxbMC44LC00Ljc5XV0sInN0cm9rZVdpZHRoIjoxLjV9LHsiaWQiOiI5MzUxYjdkNi1lMjRmLTRlYjctOTJlYi03N2IwMWY3NDNlN2UiLCJ0eXBlIjoiYmV6aWVyIiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXCh5PWYnKHgpXFwpIiwicG9pbnRzIjpbWzEuMzMsMi4zOV0sWzEuMjIsLTEuMDVdLFsxLjY5LC0yLjc4XSxbMy44LC00LjQyXV0sImxhYmVsT2Zmc2V0IjpbLTE1LDIwMF0sInN0cm9rZVdpZHRoIjoxLjV9LHsieCI6LTEuMzgsInkiOjAsImlkIjoiY2JlMzBmZWUtMjhmYi00NTE0LWFjZWYtNjE1MjE4NDg1YTBjIiwidHlwZSI6InBvaW50IiwiY29sb3IiOiIjMDAwMDAwIiwibGFiZWwiOiJcXChcXHNxcnR7a31cXCkiLCJwb2ludFNpemUiOjEsImxhYmVsT2Zmc2V0IjpbLTIwLC0xNV19LHsieCI6MS4zOCwieSI6MCwiaWQiOiJkYzBhNjJmNi1kZTJmLTQ3MzctOGFkMC00OGMxMDE3NTJlNWUiLCJ0eXBlIjoicG9pbnQiLCJjb2xvciI6IiMwMDAwMDAiLCJsYWJlbCI6IlxcKFxcc3FydHtrfVxcKSIsInBvaW50U2l6ZSI6MSwibGFiZWxPZmZzZXQiOlsxMCwtMTVdfSx7IngiOjAsInkiOi0xLjk2LCJpZCI6Ijg3ODQxNTBhLTc2Y2MtNDQzNC1hZWZmLTNjYjc1NjRlYmVmNiIsInR5cGUiOiJwb2ludCIsImNvbG9yIjoiIzAwMDAwMCIsImxhYmVsIjoiXFwoLWtcXCkiLCJwb2ludFNpemUiOjEsImxhYmVsT2Zmc2V0IjpbLTIwLC0zXX1dLCJ4QXhpc0xhYmVsIjoiXFwoeFxcKSIsInlBeGlzTGFiZWwiOiJcXCh5XFwpIiwiYm91bmRpbmdCb3giOlstMiwyLjUsNSwtNV0sIm9yaWdpbkxhYmVsIjoiXFwoT1xcKSIsImdyaWRTZXR0aW5ncyI6eyJtaW5vckVsZW1lbnRzWCI6NCwibWlub3JFbGVtZW50c1kiOjR9LCJzaG93QXhpc0xhYmVscyI6ZmFsc2UsInNob3dOYXZpZ2F0aW9uIjpmYWxzZSwia2VlcEFzcGVjdFJhdGlvIjp0cnVlLCJ4QXhpc0xhYmVsT2Zmc2V0IjpbMiwtMTFdLCJ5QXhpc0xhYmVsT2Zmc2V0IjpbLTE0LDVdLCJvcmlnaW5MYWJlbE9mZnNldCI6Wy01LC0yXX0=" data-width="400" data-height="350" style="width:400px;height:350px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li data-list-item-id="item-1761897639058-eh03s1csz" data-list-item-path="item-1761897639058-eh03s1csz"><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y = \mathrm{f}(x).$ Hence determine the nature of these stationary points.</div></li><li data-list-item-id="item-1761897639726-m4izzszvf" data-list-item-path="item-1761897639726-m4izzszvf"><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y = \mathrm{f}(x)$ is</div><ol class="i-list"><li data-list-item-id="subitem-1761897699680-4tfzaqg5e" data-list-item-path="item-1761897639726-m4izzszvf/subitem-1761897699680-4tfzaqg5e"><div class="list-item-paragraph">concave downwards,</div></li><li data-list-item-id="subitem-1761897700468-uvvzyhtgw" data-list-item-path="item-1761897639726-m4izzszvf/subitem-1761897700468-uvvzyhtgw"><div class="list-item-paragraph">decreasing.</div></li></ol></li></ol>

The diagram below shows the curve $y = \mathrm{f}'(x).$

<p>The diagram below shows the curve $y = \mathrm{f}'(x).$</p>
<div class="graph-block" data-graph-config="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" data-width="400" data-height="350" style="width:400px;height:350px;max-width:100%;display:block;margin:0 auto;"></div>
<ol class="abc-list"><li><div class="list-item-paragraph">State the $x$-coordinates of the stationary points for the curve $y = \mathrm{f}(x).$<br></div><div class="list-item-paragraph">Hence determine the nature of these stationary points.</div></li><li><div class="list-item-paragraph">State the maximal range of values of $x$ such that the curve $y = \mathrm{f}(x)$ is</div><ol class="i-list"><li><div class="list-item-paragraph">concave downwards,</div></li><li><div class="list-item-paragraph">increasing.</div></li></ol></li></ol>

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<p>Given that the coordinates of $A$ and $B$ are $ \left(a,0 \right) $ and $ \left(b,2 \right) $ respectively. <br>State the maximal range of values of $x$ for which the graph of $y = \mathrm{f}(x)$ is</p>
<ol class="abc-list"><li data-list-item-id="item-1761897094743-zoz3j0yp6" data-list-item-path="item-1761897094743-zoz3j0yp6"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1761897095912-1mjv5vka8" data-list-item-path="item-1761897095912-1mjv5vka8"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1761897096408-8g62yxqxx" data-list-item-path="item-1761897096408-8g62yxqxx"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1761897097203-q7oqq43mz" data-list-item-path="item-1761897097203-q7oqq43mz"><div class="list-item-paragraph">concave downwards.</div></li></ol>

Given that the coordinates of $A$ and $B$ are $ \left(a,0 \right) $ and $ \left(b,2 \right) $ respectively. 
State the maximal range of values of $x$ for which the graph of $y = \mathrm{f}(x)$ is

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<p>State the maximal range of values of $x$ for which the graph of $y = \mathrm{f}(x)$ is</p>
<ol class="abc-list"><li><div class="list-item-paragraph">strictly decreasing,</div></li><li><div class="list-item-paragraph">decreasing,</div></li><li><div class="list-item-paragraph">concave upwards,</div></li><li><div class="list-item-paragraph">concave downwards.</div></li></ol>

State the maximal range of values of $x$ for which the graph of $y = \mathrm{f}(x)$ is

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<p>State the maximal range of values of $x$ for which the graph of $y = \mathrm{f}(x)$ is</p>
<ol class="abc-list"><li data-list-item-id="item-1761896475367-grgkods60" data-list-item-path="item-1761896475367-grgkods60"><div class="list-item-paragraph">strictly decreasing,</div></li><li data-list-item-id="item-1761896476207-6vtpo6l8k" data-list-item-path="item-1761896476207-6vtpo6l8k"><div class="list-item-paragraph">decreasing,</div></li><li data-list-item-id="item-1761896477220-egvvma9dj" data-list-item-path="item-1761896477220-egvvma9dj"><div class="list-item-paragraph">concave upwards,</div></li><li data-list-item-id="item-1761896480992-io3cs60ap" data-list-item-path="item-1761896480992-io3cs60ap"><div class="list-item-paragraph">concave downwards.</div></li></ol>

<ol class="abc-list"><li><div class="list-item-paragraph">Without using a calculator, solve the inequality $\frac{4x}{{{x}^{2}}-3x-4}\ge -x$. Hence solve the inequality $\frac{4\left| x \right|}{{{x}^{2}}-3\left| x \right|-4}&lt;-\left| x \right|$.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">The diagrams below show the graphs of $y=\left| \mathrm{f}(x) \right|$ and $y=\mathrm{f}'(x)$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1772447541701-4on7e987hzh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzI0NDc1NDE3MDEtNG9uN2U5ODdoemgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjgzOTEwMSwiZXhwIjoxNzkyODM5NDAxfQ.PrH0JIbc76VLj4nLjrXUxOgEPKFf9e58CAmJufg-mhI" alt="List item image" class="aligncenter custom-width" style="max-width: min(500px, 100%); height: auto;" /></div><div class="list-item-paragraph">Sketch the graph of $y=\mathrm{f}(x)$, labelling the equations of any asymptotes and the coordinates of the turning point and the points where the curve crosses the axes.<span class="inline-marks">[3]</span></div></li></ol>

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