Suppose f is a real-valued function on $[−1, 1]$
f has a continuous second derivative and $f(0) = 0$.
Define a function $g$ by $g(x) = f'(0), x=0$
$g(x)=f(x)/x$ at all other x.
Prove that g is continuous and differentiable at $x = 0$ and express $g'(0)$ in terms of the derivatives of $f$ at $x = 0$.
Proving g is continuous was a matter of realizing it is equivalent to the differentiability of f at x=0. That we have, so we are done.
To know what $g'(0)$ should look like IF $g$ were differentiable, I used L'Hopital and got that it is $-f''(0)/ 2 $
I am having trouble in using the epsilon delta definition of differentiability to prove that this is indeed the correct derivative. What I have at my disposable is : since second derivative exists, $f'(x)$ is differentiable and hence continuous. Similarly, f is continuous. But I can't seem to manipulate what I have to prove that this is the correct derivative. Can someone please help me?