$F(x)=\frac{1}{x}\int_0^x f(t)dt$ is monotone if and only if $f$ is monotone
87570
Prove that if a function's average is nondecreasing then the function is nondecreasing.
158180
An inequality of $\int_0^1 |f(x)|dx$
21339
IF $f(x) = \int_{0}^{\phi (x)} g(t) dt$, How could we find $f'(x)$?
99458
Suppose $\int_{[a,b]}f=0,\text{ then }f(x)=0 \forall x\in[a,b]$
68538
Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$
142138
Show that $f(x)$ is a constant function if $\lim\limits_{h\to 0}\frac{1}{h^3}\int_{-h}^{h}f(x+t)\cdot t\,dt=0$ for all $x$
150699
Show that if $f'$ is strictly increasing, then $\frac{f(x)}{x}$ is increasing over $(0,\infty)$
68774
Proving that $f(x)$ is less than or equal to $1+\pi/4$
33523
If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing.
98836
Showing $\int_x^{x+1}f(t)\,dt \xrightarrow{x\to\infty}0$ for $f\in L^2 (\mathbb{R})$
100765
If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
35264
Must a monotone function have a monotone derivative?
53024
$f(x)f(1/x)=f(x)+f(1/x)$
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
65889
Showing $\Phi(x)=\int_0^x (x-t)f(t)\,dt$ is twice differentiable and finding $\Phi''(x)$
34398
Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$
50472
If monotone decreasing and $\int_0^\infty f(x)dx <\infty$ then $\lim\limits_{x\to\infty} xf(x)=0.$