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Solution for F(X) = 1+2 Sin X+3 Cos2x, 0 < X < 2 π 3 is (A) Minimum at X = π 2 (B) Maximum at X = Sin − 1 ( 1 √ 3 ) (C) Minimum at X = π 6 (D) Maximum at Sin − 1 ( 1 6 ) - CBSE (Science) Class 12 - Mathematics

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Question

f(x) = 1+2 sin x+3 cos2x, \[0\frac{<}{}x\frac{<}{}\frac{2\pi}{3}\] is 

(a) Minimum at x =\[\frac{\pi}{2}\]

(b) Maximum at x = sin \[- 1\] ( \[\frac{1}{\sqrt{3}}\])

(c) Minimum at x = \[\frac{\pi}{6}\]

(d) Maximum at sin \[- 1\] (\[\frac{1}{6})\]

Solution

\[(a)\text { Minimum at } x = \frac{\pi}{2}\]

\[\text { Given }: f\left( x \right) = 1 + 2 \sin x + 3 \cos^2 x\]

\[ \Rightarrow f'\left( x \right) = 2 \cos x - 6 \cos x \sin x\]

\[ \Rightarrow f'\left( x \right) = 2 \cos x\left( 1 - 3 \sin x \right)\]

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

\[f'\left( x \right) = 0\]

\[ \Rightarrow 2 \cos x\left( 1 - 3 \sin x \right) = 0\]

\[ \Rightarrow 2 \cos x = 0 or \left( 1 - 3 \sin x \right) = 0\]

\[ \Rightarrow \cos x = 0 \ or \sin x = \frac{1}{3}\]

\[ \Rightarrow x = \frac{\pi}{2} or x = \sin^{- 1} \left( \frac{1}{3} \right)\]

\[\text { Now,} \]

\[f''\left( x \right) = - 2 \sin x - 6 \cos 2x\]

\[ \Rightarrow f''\left( \frac{\pi}{2} \right) = - 2 \sin \frac{\pi}{2} - 6 \cos \left( 2 \times \frac{\pi}{2} \right) = - 2 + 6 = 4 > 0\]

\[\text { So, x } = \frac{\pi}{2} \text { is a local minima }.\]

\[\text { Also }, \]

\[f''\left( \sin^{- 1} \left( \frac{1}{3} \right) \right) = - 2 \sin \left( \sin^{- 1} \left( \frac{1}{3} \right) \right) - 6 \cos \left( \sin^{- 1} \left( \frac{1}{3} \right) \right) = \frac{- 2}{3} - 6 \times \frac{2\sqrt{2}}{3} = - \left( \frac{2}{3} + 4\sqrt{2} \right) < 0\]

\[\text { So,} x = \sin^{- 1} \left( \frac{1}{3} \right)\text {  is a local maxima }.\]

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Solution for question: F(X) = 1+2 Sin X+3 Cos2x, 0 < X < 2 π 3 is (A) Minimum at X = π 2 (B) Maximum at X = Sin − 1 ( 1 √ 3 ) (C) Minimum at X = π 6 (D) Maximum at Sin − 1 ( 1 6 ) concept: Graph of Maxima and Minima. For the courses CBSE (Science), PUC Karnataka Science, CBSE (Arts), CBSE (Commerce)
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