Let $\ f:]a,b[\rightarrow\mathbb{R}$ be a differentiable function on $]a,b[$ so that :
$\lim_{x\rightarrow a}_{>}\ f(x) = \lim_{x\rightarrow b}_{<}\ f(x) = +\infty$
Show that there exists $c\in ]a,b[$ so that $\ f'(c) = 0$.
What I did is that i separated this problem in three cases. Let $\delta>0$ and consider $a_1 = a+\delta$ and $b_1 = b-\delta$. Now we have the following cases: $f(a_1)=f(b_1)$, $\ f(a_1)
When $f(a_1)=f(b_1)$ we can directly apply Rolle's theorem on $]a_1,b_1[$ so $\exists c\in ]a_1,b_1[ $ such that $f'(c)=0$.
Now I'm not sure about the case where $\ f(a_1)
The last case is analogous to the previous one.
Could someone tell me if the proof is correct, especially for the case where $\ f(a_1)
Thank you in advance.