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<p>The diagram below shows the curve $y=\mathrm{{f}'}\left( x \right).$ It has turning points at $\left( 0,6 \right),\left( 5,0 \right)$ and $\left( 7,7 \right)$ and intersects the $x$-axis at $\left( -3,0 \right)$ and $\left( 8,0 \right).$</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/converted/1759146991968-xhasbm72jl.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvY29udmVydGVkLzE3NTkxNDY5OTE5NjgteGhhc2JtNzJqbC5qcGciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyODM5MTkwLCJleHAiOjE3OTI4Mzk0OTB9.YVphFcm07dq9e9dL3aOS7i9iRxiSn2UALvlZTfnbqLs" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li data-list-item-id="item-1759146991479-g6uwv70af" data-list-item-path="item-1759146991479-g6uwv70af"><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.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-1759146992454-1fcqbj8nr" data-list-item-path="item-1759146992454-1fcqbj8nr"><div class="list-item-paragraph">State the range of values of $x$ such that the curve $y=\mathrm{f}\left( x \right)$ is concave downwards.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1759146993616-lq1l3r4vs" data-list-item-path="item-1759146993616-lq1l3r4vs"><div class="list-item-paragraph">A student makes a claim on each of the following statements:</div><ol class="abc-list"><li data-list-item-id="subitem-1759147015862-f3e93cz39" data-list-item-path="item-1759146993616-lq1l3r4vs/subitem-1759147015862-f3e93cz39"><div class="list-item-paragraph">If the curve $y=\mathrm{f}\left( x \right)$ passes through the point $\left( -3,7 \right),$ then&nbsp;$\mathrm{f}\left( x \right)&gt;7$ for&nbsp;$-3 &lt; x &lt; 5.$</div></li><li data-list-item-id="subitem-1759147016619-jnundrs0l" data-list-item-path="item-1759146993616-lq1l3r4vs/subitem-1759147016619-jnundrs0l"><div class="list-item-content"><span class="list-item-text">By using integration by parts, $\displaystyle\displaystyle \displaystyle\int_{-3}^{8}{x\mathrm{f}\left( x \right)}\,\mathrm{d}x&gt;32\mathrm{f}\left( 8 \right)-\frac{9}{2}\mathrm{f}\left( -3 \right).$</span></div><div class="list-item-paragraph">For each of the above statements, explain briefly whether the student is right to make such a claim.<span class="inline-marks">[3]</span></div></li></ol><div class="list-item-after-content">For each of the above statements, explain briefly whether the student is right to make such a claim.<span class="inline-marks">[3]</span></div></li></ol>

The diagram below shows the curve $y=\mathrm{{f}'}\left( x \right).$ It has turning points at $\left( 0,6 \right),\left( 5,0 \right)$ and $\left( 7,7 \right)$ and intersects the $x$-axis at $\left( -3,0 \right)$ and $\left( 8,0 \right).$

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.

State the range of values of $x$ such that the curve $y=\mathrm{f}\left( x \right)$ is concave downwards.

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