Question

The total number of local maxima and local minima of the function $f\left(x\right)={(2+x)}^3$, when $-3<x\le -1$ and $x^{\frac{2}{3}}$, when $-1<x<2$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C

$2$

Total no of local maxima and local minima:

Given , $f\left(x\right)=\left\{\begin{array}{ll}{(2+x)}^3& -3<x\le -1\\ x^{\frac{2}{3}}& -1<x<2\end{array}\right.$

Differentiate with respect to $x$,

$f\text{'}\left(x\right)=\left\{\begin{array}{ll}3{(2+x)}^2& -3<x\le -1\\ \frac{2}{3}{x}^{-\frac{1}{3}}& -1<x<2\end{array}\right.$

For $-3<x\le -1,f\text{'}\left(x\right)>0$ , function is increasing, so no local maxima and minima under this range.

For $-1<x<0,f\text{'}\left(x\right)<0$ and for $0<x<2,f\text{'}\left(x\right)>0$

Therefore, one local maximum at $x=-1$ and one local minimum at $x=0$.

Thus, there are total number of $2$ local maxima and local minima.

Hence option (c) is the correct option.