Question

Discuss the applicability of Rolle's theorem for the following function on the indicated interval:
$f\left(x\right)=x^{2/3}$ on $[-1,1]$

Answer 1

Given the function $f\left(x\right)=x^{2/3}$ on $[-1,1]$.

Now, $f(-1)=1=f\left(1\right)$.

Also the function is continuous in $[-1,1]$.

But the given function is not differentiable at $x=0$.

Since,

$\underset{h\to 0}{lim}\frac{h^{\frac{2}{3}}-0}{h}$

$=\underset{h\to 0}{lim}\frac{1}{h^{\frac{1}{3}}}$ does not exist.

So this function does not satisfy the Rolle's theorem on $[-1,1]$.