$f$ twice continuously differentiable on $[a, b], f^\prime(c) = 0$ for some $c \in[a, b], f''(c) > 0$ implies various statements

Question

Assume $f$ is twice continuously differentiable on $[a, b]$ and there's some $c \in [a, b]$ such that $f'(c) = 0$. Suppose $f''(c) > 0$. Prove that:

1. there exists $\delta > 0$ such that $f''(x) > 0$ for all $x \in (c - \delta, c + \delta)$.

(I was able to prove this already, but I'm leaving it here in case it's helpful for 2. and 3., which I haven't had any luck with.)

2. $f'(x) < 0$ for $x \in (c - \delta, c)$ and $f'(x) > 0$ for $x \in (c, c + \delta)$.

What I'm confused about here is finding a way to apply $\epsilon-\delta$ continuity (if that is even the right way to approach this part).

3. $f(x) > f(c)$ for $x \in (c - \delta, c + \delta)$ such that $x \neq c$.

Thanks in advance.

Answer 1

Hint:

By the mean value theorem since $f'$ is differentiable there exists $\xi$ between $x$ and $c$ such that

$$f'(x) = f'(c) + f''(\xi)(x-c) = f''(\xi)(x-c),$$

and, by what you have already shown, $f''(\xi) > 0$ for any $\xi \in (c-\delta,c+\delta)$

Answer 2

There is a counter example to 1) Take $f(x)= x^2sin(1/x)+ \frac{x}{2} \forall x\in(-\infty,0)\cup(0,+\infty)$ and $f(x)=0$ for $x=0$. Take a look

Answer 3

f'' is continious .So $\forall ε>0 \exists δ(ε)>0 : \forall x\in (c-δ,c+δ)$

$|f''(x)-f''(c)|<ε,$

So se can we choose $ ε=ε_0<f''(c)$

Τhen $\exists δ(ε_0)>0 : \forall x\in (c-δ(ε_0),c+δ(ε_0))$

$|f''(x)-f''(c)|<ε_0$

So $0<f(c)-ε_0<f''(x)<f(c)+ε_0,\forall x\in (c-δ(ε_0),c+δ(ε_0))$

Αnd we are done for 1)

Answer 4

For part c, I first used the Mean Value Theorem that: for every interval $(c-\delta, c)$, we can find an $x_1$ and (likewise and $x_2$ for the interval $(c, c+\delta)$), such that: $$f'(x_1) = (f(c) - f(c-\delta))/\delta$$ and $$f'(x_2) = (f(c+\delta) - f(c))/\delta$$

But we know from part b that $f'(x_1) <0 \implies f(c-\delta) > f(c)$ and $f'(x_2) >0 \implies f(c+\delta) > f(c)$ .

(*) Now we can choose smaller intervals within given interval, e.g. ($c-\delta_i, c+\delta_i$), where $\delta_i < \delta $, ($\delta_i>0$), and attain all possible values of f(x) = $f(c-\delta_i)>f(c)$ or $ f(c+\delta_i) >f(c)$ $ \implies f(x) >$ f(c).

I'm a little unsure whether my argument (*) works. I'm saying that for all x in the given interval, $f(x)> f(c)$ because any x in that interval can be found to equal a $c+\delta_i$ or $c-\delta_i$, which we found from part b is greater than f(c).