#### Provide solution RD Sharma maths class 12 chapter maxima and minima exercise 17.2 question 9 maths textbook solution

There  is no   local maxima and  local minima of $f\left ( x \right )$ at interval (0,π)

Hint:

Use first derivative test to find the point and value of local maxima or local minima.

Given:

$f(x)=\operatorname{Cos} x \quad 0

Solution:

$f(x)=\operatorname{Cos} x$

Differentiating  $f\left ( x \right )$with respect to ‘x’ then,

\begin{aligned} &\frac{d}{d x}\{f(x)\}=\frac{d}{d x} \operatorname{Cos} x \\ &\because f^{\prime}(x)=-\sin x\left[\because \frac{d(\cos x)}{d x}=-\sin x\right] \end{aligned}

By first derivative test, for local maxima or local minima ,we have

\begin{aligned} &f^{\prime}(x)=0 \\ &\Rightarrow-\sin x=0 \Rightarrow \operatorname{Sin} x=0 \\ &\Rightarrow x=n \pi \quad ; \mathrm{n} \in \mathbb{Z} \\ &\Rightarrow \mathrm{x}=0, \pi,-\pi, 2 \pi,-2 \pi \ldots . \end{aligned}

But these points of x lies outside the interval (0,π)

So there is no local maxima and minima will exist in the interval (0,π)