Is this function differentiable at zero?
$\displaystyle f(x)=\mid x\mid ^\frac{5}{2}$
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 $\displaystyle \frac{f(x)}{x}$ has a limit as $\displaystyle x\to 0$. In the case of $\displaystyle f(x)=|x|^\alpha$, the ratio equals $\displaystyle |x|^{\alpha-1}$ if $\displaystyle x>0$ and $\displaystyle -|x|^{\alpha-1}$ if $\displaystyle x<0$ (because then $\displaystyle |x|=-x$), hence
- it converges to 0 if $\displaystyle \alpha>1$;
- if $\displaystyle \alpha=1$, it converges to 1 from the right and -1 from the left, hence it doesn't converge
- if $\displaystyle 0<\alpha<1$, it diverges to $\displaystyle +\infty$ from the right and $\displaystyle -\infty$ from the left, hence it doesn't converge.
As a conclusion, it is differentiable iff $\displaystyle \alpha>1$, and the derivative is 0 then.