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please solve rd sharma class 12 chapter 16 Increasing and Decreasing Functions exercise Multiple choice question , question 6 maths textbook solution

Answers (1)

Correct option (c)

Hint: If f(x)  is increasing function {f}'(x)>0

Given: f(x)=x^{3}+a x^{2}+b x+5 \sin ^{2} x

Explanation: It is given that

 f(x)=x^{3}+a x^{2}+b x+5 \sin ^{2} x

Differentiate w.r.t  x

\begin{aligned} f^{\prime}(x) &=3 x^{2}+2 a x+b+5 \cdot \sin x \cdot \cos x \\ &=3 x^{2}+2 a x+b+5 \cdot \sin 2 x \quad[\because \sin 2 x=2 \sin x \cos x] \end{aligned}

\because If f(x)  is increasing  {f}'(x)>0

\therefore 3 x^{2}+2 a x+b+5 \cdot \sin 2 x>0

For the quadratic equation

Discriminant is

 \begin{aligned} &(2 a)^{2}-4 \times 3(b+5 \sin 2 x)<0 \\ &4 a^{2}-12 b-60 \sin 2 x<0 \\ &a^{2}-3 b-15 \sin 2 x<0 \end{aligned}

\because Minimum value of \sin 2x=-1

So, a^{2}-3 b-15(-1)<0

Thus, a^{2}-3 b+15<0

So, a & b satisfy equation  a^{2}-3 b+15<0

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