First slide

Extreme values and Periodicity of Trigonometric functions

Question

The greatest value of $f (x) = 2 \sin x + \sin 2 x$ on $[0, 3 π / 2]$ , is given b

Moderate

A

$\frac{9}{2}$
B

$\frac{5}{2}$
C

$\frac{3 \sqrt{3}}{2}$
D

$\frac{3}{2}$

Solution

$f^{'} (x) = 2 \cos x + 2 \cos 2 x$ and $f^{''} (x) = - 2 \sin x - 4 \sin 2 x$
For extreme value $f^{'} (x) = 0$
$\begin{array}{l} \Rightarrow & \cos x + 2 \cos^{2} x - 1 = 0 \end{array}$
$\begin{array}{l} \Rightarrow & \cos x = - 1 \end{array}$ or $1 / 2$
$\begin{array}{l} \Rightarrow & x = π \end{array}$ or $π / 3$ as $x \in [0, 3 π / 2$
Now $f (π) = 0$ and $f (π / 3) = \frac{2 \sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \frac{3 \sqrt{3}}{2}$
Also $f (0) = 0$ and $f (3 π / 2) = - 2$ so the greatest value of $f (x)$ is $\frac{3 \sqrt{3}}{2}$

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