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At X= 5 π 6 F(X) = 2 Sin 3x + 3 Cos 3x is (A) 0 (B) Maximum (C) Minimum (D) None of These - Mathematics

Question

At x= $\frac{5\pi}{6}$ f(x) = 2 sin 3x + 3 cos 3x is ______________ .

Options
• 0

• maximum

• minimum

• none of these

Solution

none of these

$\text { Given }: f\left( x \right) = 2 \sin 3x + 3 \cos 3x$

$\Rightarrow f'\left( x \right) = 6 \cos 3x - 9 \sin 3x$

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

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

$\Rightarrow 6 \cos 3x - 9 \sin 3x = 0$

$\Rightarrow 6 \cos 3x = 9 \sin 3x$

$\Rightarrow \frac{\sin 3x}{\cos 3x} = \frac{2}{3}$

$\Rightarrow \tan 3x = \frac{2}{3} . . . \left( 1 \right)$

$\text { At x } = \frac{5\pi}{6}:$

$\tan 3x = \tan \frac{5\pi}{2}$

$\Rightarrow \tan 3x = \tan \frac{\pi}{2}$

$\text { So,} \tan 3x \text { is not defined }. \left[ \tan 3x \neq \frac{2}{3} \text { is not satisfying eq } . \left( 1 \right) \right]$

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

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