<p>Let&nbsp;\(f(x) = 3 \sin{x} + 4\cos{x}\), for&nbsp;\(-2\pi \leq x \leq 2\pi\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of \(f\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down</div><ol class="i-list"><li><div class="list-item-paragraph">the amplitude;</div></li><li><div class="list-item-paragraph">the period;</div></li><li><div class="list-item-paragraph">the \(x\)-intercept that lies between \(-\frac{\pi}{2}\) and \(0\).<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph">Hence write \(f(x)\) in the form \(p \sin{(qx + r)}\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Write down one value of \(x\) such that&nbsp;\(f'(x) = 0\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Write down the two values of \(k\) for which the equation \(f(x) = k\) has exactly two solutions.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Let&nbsp;\(g(x) = \ln{(x +1)}\), for&nbsp;\(0 \leq x \leq \pi\). There is a value of \(x\), between \(0\) and \(1\), for which the gradient of \(f\) is equal to the gradient of \(g\). Find this value of \(x\).<span class="inline-marks">[5]</span></div></li></ol>

Let \(f(x) = 3 \sin{x} + 4\cos{x}\), for \(-2\pi \leq x \leq 2\pi\).

Hence write \(f(x)\) in the form \(p \sin{(qx + r)}\).

Write down one value of \(x\) such that \(f'(x) = 0\).

Write down the two values of \(k\) for which the equation \(f(x) = k\) has exactly two solutions.

Let \(g(x) = \ln{(x +1)}\), for \(0 \leq x \leq \pi\). There is a value of \(x\), between \(0\) and \(1\), for which the gradient of \(f\) is equal to the gradient of \(g\). Find this value of \(x\).

Trigonometric Functions and Graphs - General

Trigonometry

International Baccalaureate Mathematics: Applications and Interpretation Standard Level