Maxima and Minima — Maths STD 12 Science — Question

4. none of these.

Solution:

Given, $\text{f}(\text{x})=2\sin3\text{x}+3 \cos3\text{x}$

$\Rightarrow \text{f}'(\text{x})=6 \cos3\text{x}-9\cos3\text{x}$

to find maxima or minima f'(x) = 0

$6 \cos3\text{x}-9\cos3\text{x}=0$

$\Rightarrow \tan3\text{x}=\frac{2}{3}$

$\text{f}'\Big(\frac{5\pi}{6}\Big)=\tan\Big(3\times\frac{5\pi}{6}\Big)$

$\text{f}'\Big(\frac{5\pi}{6}\Big)=\tan\Big(\frac{5\pi}{2}\Big)$

$\Rightarrow\text{f}'\Big(\frac{5\pi}{6}\Big)=\tan\Big(2\pi+\frac{\pi}{2}\Big)$

$\Rightarrow\text{f}'\Big(\frac{5\pi}{6}\Big)=\tan\Big(\frac{\pi}{2}\Big)$ which is not defined.

Hence, $\text{x}=\frac{5\pi}{6}$ is not a critical point.