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Maxima and Minima — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMaxima and Minima1 Mark
Question
At $\text{x}=\frac{5\pi}{6}, $ $\text{f}(\text{x})=2\sin3\text{x}+3 \cos3\text{x}$ is:
  1. 0
  2. maximum.
  3. minimum.
  4. none of these.
✓

Answer

  1. 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.

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