# differentiability

• February 20th 2011, 08:34 AM
Sambit
differentiability
Let $f:R-->R$ be a function defined as
$f(x)=x^{3/2}$,x>=0
$-|x|^{3/2}$, x<0.

What can be said about its differentiability? Is is differentiable at 0? If yes, how many times?
• February 20th 2011, 09:09 AM
FernandoRevilla
Verify $f'_{+}(0)=f'_{-}(0)=0$ so , :

$f'(x)=\begin{Bmatrix}{ 3\sqrt{x}/2}&\mbox{ if }& x>0\\0 & \mbox{if}& x=0\\\ldots & \mbox{if}& x<0\end{matrix}$

Now, you can study the existence or not of $f''(0)$ .

Fernando Revilla
• February 20th 2011, 06:54 PM
Sambit
I am getting $f(x)=x^{3/2}$,x>=0
$-x^{3/2}$,x<0
which is differentiable.

Differentiating, $f'(x)=\frac{3}{2}x^{1/2}$, x>=0
$\frac{3}{2}(-x)^{1/2}$,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?
• February 21st 2011, 12:33 AM
FernandoRevilla
Quote:

Originally Posted by Sambit
So I think it is correct that the function is infinitely differentiable. Am I right?

No, prove that:

$\displaystyle\lim_{h \to 0^+}{\dfrac{f'(h)-f'(0)}{h}}=\ldots=+\infty$

so, $f''(0)$ does not exist.

Fernando Revilla
• February 21st 2011, 04:17 AM
Sambit
Okay....got it. thank you.