[Math] Why doesn’t this contradict Rolle’s theorem

Question

So I graphed $y=5/2x^{2/5}$:

So according to Rolle's theorem,

$(f(1)-f(-1))/(1-(-1))=(5/2-5/2)/2=0$

And the first derivative is $1/x^{3/5}$.

But, there's no way $1/x^{3/5}=0$. What's going on here? Taking a look at the graph, it looks like its slope never hits zero anyways, so I'm sure there might be a restriction for rolle's theorem, but I'm unsure what.

Answer 1

Rolle's theorem is stated as follows:

Rolle's Theorem. If $f:[a,b]\to\Bbb R$ is continuous and $f$ is differentiable on $(a,b)$ with $f(a)=f(b)$, then there exists a $c\in(a,b)$ such that $f^\prime(c)=0$.

The problem with the function $f(x)=5/2x^{2/5}$ is that it is not differentiable at $x=0$ so Rolle's theorem does not apply on any interval containing $0$.