PUC Karnataka Science Class 12Department of Pre-University Education, Karnataka
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# Solution for F(X) = Sin + √ 3 Cos X is Maximum When X = (A) π 3 (B) π 4 (C) π 6 (D) 0 - PUC Karnataka Science Class 12 - Mathematics

#### Question

f(x) = $\sin + \sqrt{3} \cos x$ is maximum when x =

(a) $\frac{\pi}{3}$

(b) $\frac{\pi}{4}$

(c) $\frac{\pi}{6}$

(d) 0

#### Solution

$(c) \frac{\pi}{6}$

$\text { Given }: f\left( x \right) = \sin x + \sqrt{3} \cos x$

$\Rightarrow f'\left( x \right) = \cos x - \sqrt{3} \sin x$

$\text { For a local maxima or a local minima, we must have }$

$f'\left( x \right) = 0$

$\Rightarrow \cos x - \sqrt{3} \sin x = 0$

$\Rightarrow \cos x = \sqrt{3} \sin x$

$\Rightarrow \tan x = \frac{1}{\sqrt{3}}$

$\Rightarrow x = \frac{\pi}{6}$

$\text { Now,}$

$f''\left( x \right) = - \sin x - \sqrt{3} \cos x$

$\Rightarrow \Rightarrow f''\left( \frac{\pi}{2} \right) = - \sin\frac{\pi}{2} - \sqrt{3} \cos\frac{\pi}{2}\frac{- 1}{2} - \frac{3}{2} = - 2 < 0$

$\text { So,} x = \frac{\pi}{2}\text { is a local maxima }.$

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Solution F(X) = Sin + √ 3 Cos X is Maximum When X = (A) π 3 (B) π 4 (C) π 6 (D) 0 Concept: Graph of Maxima and Minima.
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