if $f'(a)<0<f'(b)$ then there exists $c$ such that $f'(c) = 0$
53767
How to prove that derivatives have the Intermediate Value Property
88311
Linear functional f(v)=0 imply v=0
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
94944
Differentiable functions satisfying $f'(f(x))=f(f'(x))$
113892
Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
18905
$f(a)=f(b)=f_{+}'(a)=f_{-}'(b)=0$ and $|f''(x)|\le M$. Show that $|f(x)|\le \frac{M(b-a)^2}{16}$.
129716
Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
100765
If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
16212
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) d(x) = 0$ iff $f(x) = 0$
105084
Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
21879
Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
17154
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
111186
Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
149399
Given continuous $f$ and $g$, there exists $c \in (a,b)$ such that $\int_{a}^{c} f(x) \ \mathrm dx = \int_{c}^{b} g(x) \ \mathrm dx$
31522
Proving $f$ has at least one zero inside unit disk