Question

Let f:R → R be a continuous function that is convex on (-infinity,0] and [0, infinity) and has a local maximum at the point 0. Prove that the function f is not differentiable at the point 0.​

Answer 1

Answer:

See attachment.

Step-by-step explanation:

We can solve for LHD and RHD separately if f(x) is given.

Answer 2

Let, $f:R$,

where, as R is the continuous function,

as we have to prove that function f is not differentiable at 0,

As suppose f is twice differentiable,

and $f'(x)=0 at x=0$

Since, at $x = 0$,

then, $f''(x)<0$

Now, f is convex on $f''(x)>0$ in that interval,

So, $f'(x)$ doesn't exist at $x=0$