Let $\displaystyle f:R-->R$ be a function defined as
$\displaystyle f(x)=x^{3/2}$,x>=0
$\displaystyle -|x|^{3/2}$, x<0.
What can be said about its differentiability? Is is differentiable at 0? If yes, how many times?
Verify $\displaystyle f'_{+}(0)=f'_{-}(0)=0$ so , :
$\displaystyle f'(x)=\begin{Bmatrix}{ 3\sqrt{x}/2}&\mbox{ if }& x>0\\0 & \mbox{if}& x=0\\\ldots & \mbox{if}& x<0\end{matrix} $
Now, you can study the existence or not of $\displaystyle f''(0)$ .
Fernando Revilla
I am getting $\displaystyle f(x)=x^{3/2}$,x>=0
$\displaystyle -x^{3/2}$,x<0
which is differentiable.
Differentiating, $\displaystyle f'(x)=\frac{3}{2}x^{1/2}$, x>=0
$\displaystyle \frac{3}{2}(-x)^{1/2}$,x<0
This is also continuous and differentiable at 0, and so on. So I think it is correct that the function is infinitely differentiable. Am I right?
No, prove that:
$\displaystyle \displaystyle\lim_{h \to 0^+}{\dfrac{f'(h)-f'(0)}{h}}=\ldots=+\infty$
so, $\displaystyle f''(0)$ does not exist.
Fernando Revilla