If $|f'(c)|<M$, prove $|\int_{0}^{1}f(x)dx-1/n \sum_{k=0}^{n-1}f(x/n)|<M/n$
34136
$f$ is differentiable in $[0,1]$ ,$\sup_{x\in[0,1]}|f'(x)| \le M\lt+\infty $
149586
If $\lim_{x\to\infty} f'(x) = L$, prove that $\lim_{x\to\infty} f(x+c)-f(x) = cL$ and $\lim_{x\to\infty} f(x)/x=L$
20089
Prove that $\lim_{N\rightarrow\infty}(1/N)\sum_{n=1}^N f(nx)=\int_{0}^1f(t)dt$
46844
Show that if $\int_0^x f(y)dy \sim Ax^\alpha$ then $f(x)\sim \alpha Ax^{\alpha -1}$
78174
How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
33097
Proving an inequality without an integral: $\frac {1}{x+1}\leq \ln (1+x)- \ln (x) \leq \frac {1}{x}$
97558
Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration
16418
Showing that $\frac{f(x)-f(x-h)}{h} \le f'(x) \le \frac{f(x+h)-f(x)}{h}$ given that f' is increasing
78834
if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
133990
How prove this $\int_{a}^{b}f(x)dx=\frac{1}{2}(b-a)[f(a)+f(b)]-\frac{1}{12}(b-a)^3f''(\xi)$
33617
Prove the inequality involving multiple integrals
27705
Proof of $\int_{[0,\infty)}pt^{p-1}\mu(\{x:|f(x)|\geq t\})d\mu(t)=\int_{[0,\infty)}\mu(\{x:|f(x)|^p\geq s\})d\mu(s)$
98344
Prove that $\int_0^1[f''(x)]^2dx\ge4.$
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
53001
If $|f'(c)| \leq M$ and $ M < 1$, the sequence defined by $a_{n+1} = f(a_n)$ converges
114184
Show: $\sum_{k=1}^{\infty}\mu(\left\{f\geq k\right\})\leq\int f\, d\mu\leq\sum_{k=0}^{\infty}\mu(\left\{f>k\right\})$