Show $g''=f$ on $[0, 1]$ with $g(x):=\int_{0}^{x}(x-t)f(t)dt$.
65889
Showing $\Phi(x)=\int_0^x (x-t)f(t)\,dt$ is twice differentiable and finding $\Phi''(x)$
109561
Find $F'(x)$ if $F(x) = \displaystyle \int_{0}^x xf(t)dt$.
133682
Prove that if $f(t)$ satisfies $f(t)=f'(t)$, then $f(t+s)=f(t)f(s)$ for all $t,s\in R$ without using the fact that $f(t)=c\cdot e^t$.
153612
Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$
110368
Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
21879
Show that if $\lim_{x\to a}f'(x) = A$ then $f'(a)$ exists and equals A
27705
Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
37015
Prove $\int_X |f|^p=p\int^{\infty}_{0} t^{p-1}\mu({x: |f(x)>t}) dt\,$
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
126461
Show that $g:I\to\mathbb{R}$ is continuous and $f:(x-x_0)g(x)$ is differentiable
98344
Prove that $\int_0^1[f''(x)]^2dx\ge4.$
67565
If $\int^x_0 f (t) dt =x+ \int^1_x t f (t) dt$, find value of $f(1)$
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
157426
Proving that:$\int_X f_n g \, d\mu \to \int_X fg \, d\mu$ for all $g$ in $\mathscr{L}^q (X)$
49408
Proof of the derivative of ln(x)
84413
Why is $\int_{-\infty}^\infty |g(x)|\,dx = \int_0^\infty \mu(\{x : |g(x)| \ge t\})\,dt$ true?
43956
Differentiating $f(x)^{g(x)}$