Is this function differentiable at zero?

Quote:

Originally Posted by featherbox

Is this function differentiable at zero?

$f(x)=\mid x\mid ^\frac{5}{2}$

I'm having a bit of trouble deciding whether functions like this are differentiable so steps would be helpful.
Thanks.

Since the value at 0 is 0, this function is differentiable at 0 if and only if $\frac{f(x)}{x}$ has a limit as $x\to 0$ . In the case of $f(x)=|x|^\alpha$ , the ratio equals $|x|^{\alpha-1}$ if $x>0$ and $-|x|^{\alpha-1}$ if $x<0$ (because then $|x|=-x$ ), hence
- it converges to 0 if $\alpha>1$ ;
- if $\alpha=1$ , it converges to 1 from the right and -1 from the left, hence it doesn't converge
- if $0<\alpha<1$ , it diverges to $+\infty$ from the right and $-\infty$ from the left, hence it doesn't converge.

As a conclusion, it is differentiable iff $\alpha>1$ , and the derivative is 0 then.