QJEEAdv08Mac86b: JEE Advanced 2008 Mathematics - Class 12 JEEAdv08Mac86b Solution: The total number of local maxima and local minima of the…

Given that, $$f(x)=\{\begin{array}{cc}{(2+x)}^3,& -3<x\le -1\\ x^{2/3},& -1<x<2\end{array}$$

$$f^{\prime }(x)=\{\begin{array}{cc}3{(2+x)}^2,& -3<x<-1\\ \frac{2}{3}{x}^{\frac{-1}{3}},& -1<x<2\end{array}$$

Clearly, $f'(x)$ changes signs from positive to negative as x passes through x $=-1$

f'(x) &lt; 0 for all $x\in(-3,-1)$

Also, f'(x)&gt;0 for all $x\in$

f'(x) &lt; 0 for all $x\in(-1,0)$

But $f'(0)$ does not exist

So, $f(x)$ attains a local minimum at x = 0

Hence, the total number of local maxima and local minima is 2.