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  • Explain solution rd sharma class 12 chapter 16 Increasing and decreasing function exercise multiple choice question, question 33

Explain solution rd sharma class 12 chapter 16 Increasing and decreasing function exercise multiple choice question, question 33

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decreasing in \left ( \frac{\pi}{2},\pi \right )

Hints: Find the derivative of given equation.

Given: f(x)=4 \sin ^{3} x-6 \sin ^{2} x+12 \sin x+100

Solution: We have,

               f(x)=4 \sin ^{3} x-6 \sin ^{2} x+12 \sin x+100

               \begin{aligned} f^{\prime}(x) &=12 \sin ^{2} x \cos x-12 \sin x \cos x+12 \cos x \\ &=12 \cos x\left[\sin ^{2} x-\sin x+1\right] \\ &=12 \cos x\left[\sin ^{2} x+(1-\sin x)\right] \end{aligned}

                         

Now 1-\sin x \geq 0 and \sin^{2} x \geq 0

               \sin ^{2} x+1-\sin x \geq 0

Hence, f'{x}>0 when \cos x>0 , i.e., x\epsilon \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )

So, f(x) is increasing when x\epsilon \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right )

f'{x}<0 , when \cos x>0 , i.e., x\epsilon \left ( \frac{\pi}{2}, \frac{3\pi}{2} \right )

Hence, f(x) is decreasing when x\epsilon \left ( \frac{\pi}{2}, \frac{3\pi}{2} \right )

f(x) is decreasing in x\epsilon \left ( \frac{\pi}{2},\pi \right )

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