Question

Find the local maxima and local minima for the given function and also find the local maximum and local minimum values $f\left(x\right)=x^3,x\in (-2,2)$

Answer 1

Given: $f\left(x\right)=x^3$

$f^{{}^{\prime }}\left(x\right)=3x^2$

Putting this equal to zero, we get

$f^{{}^{\prime }}\left(x\right)=0$ at $x=0$

Now lets look at second derivative:

$f^{{}^{\prime \prime }}\left(x\right)=6x$

At $x=0$

$f^{{}^{\prime \prime }}\left(0\right)=6\times 0$

This is also $0$ at $x=0$ which means given function will not have a maxima or minima in given domain. It will just have a point of inflection at $x=0$