$f(0)=0$,$f(1)=1$,find the largest $u$ such that $\int_0^1|f'(x)-f(x)|dx \geqslant u$.
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
107795
If $\,f''(x) \ge f(x)$, for all $x\in[0,\infty),$ and $\,f(0)=f'(0)=1$, then is $\,f(x)>0$?
67565
If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$
30860
Prove that $f(x)=0$ in $[-1,1]$.
4656
Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions
72660
How can I show that $f$ must be zero if $\int fg$ is always zero?
137301
To find the minimum of $\int_0^1 (f''(x))^2dx$
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
53748
Find maximum value of $\int_{0}^{1}\left(f(x)\right)^3dx$
5229
Prove that $\lim_{n \to \infty} \int_0^1{nx^nf(x)}dx$ is equal to $f(1)$.
21301
If nonnegative $f: [0,1] \rightarrow \mathbb{R}$ has a continuous $f''$, then $\int_0^1 \Big| \frac{f''(x)}{f(x)} \Big| \,dx >4$
145573
How to show that $f$ is a zero function?
17154
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
59063
$f(f(x))=f(x)$ question
33989
If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
91775
Show that $\int_0^1 f^2(x)dx\geq 3$, if $f:[0,1]\to \mathbb{R}$ is an integrable function s.t. $\int_0^1 f(x)dx=\int_0^1 xf(x)dx=1$