At x=5 6 , f(x)=2 3x+3 3x is: - Vidyadip

MCQ

At $\text{x}=\frac{5\pi}{6}, $ $\text{f}(\text{x})=2\sin3\text{x}+3 \cos3\text{x}$ is:

A
$0$
B
maximum.
C
minimum.
✓
none of these.

✓

Answer

Correct option: D.

none of these.

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.

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