How prove this $f'(c)=2c(f(c)-f(0))$
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
81921
If $f(x + y) = f(x) + f(y)$ showing that $f(cx) = cf(x)$ holds for rational $c$
64485
Proving $\displaystyle\lim_{h\to0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}=f''(x)$
26219
Suppose f is a function such that f'(x) = 1/x for all x > 0. Prove that if f(1) = 0, then f(xy) = f(x) + f(y) for all x,y > 0
1057
How prove this integral limit $=f(\frac{1}{2})$
101778
Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
89429
Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
18905
$f(a)=f(b)=f_{+}'(a)=f_{-}'(b)=0$ and $|f''(x)|\le M$. Show that $|f(x)|\le \frac{M(b-a)^2}{16}$.
74100
How do you solve $f'(x) = f(f(x))$?
76148
Show that $\exists c \in \mathbb{R} : f(c) = 0$
15262
Prove $\exists \xi, \eta \in (0,1)$, such that $f(\xi)+f'(\eta)=\xi+\eta$.
56967
Show that $ f^2 − g^2 = C $ for some real constant C
8854
Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$
78319
Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
24553
$f$ has the form $f(x) =ax^2+bx+c$.
57026
$f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
78834
if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $