$f(x)$ is continuously differentiable on $[0,1]$, prove that $ (\int_0^1 f(x)dx)^2 \leq \frac {1} {12} \int_0^1 (f'(x))^2 dx $
78174
How prove this inequality $\left(\int_{0}^{1}f(x)dx\right)^2\le\frac{1}{12}\int_{0}^{1}|f'(x)|^2dx$
85623
Proof of continuity implied by differentiability
31685
Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
15727
show that if $f$ is uniformly differentiable then prove that $f'$ is continuous
57695
Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.
43956
Differentiating $f(x)^{g(x)}$
111458
Prove that $f$ is differentiable
34081
Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$
129324
Let $f:[0;1]\to\mathbb R$ be continiously differentiable
136714
Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$.
53726
Prove that if $f$ is a real continuous function such that $|f|\le 1$ then $|\int_{|z|=1} f(z)dz| \le 4$
75932
Differentiability of $g:=f(\sqrt{x^2+y^2})$ for a $C^1$ function with $f'(0)=0$ NBHM $2008$
3518
Proving $f(x)=1/x$ on $(0,1 )$ is not uniformly continuous
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
20192
Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
54403
Given $f : \Bbb{R} \to \Bbb{R}$ is a continuous function. To prove that $\int_{0}^{1}f(x)x^2 dx = \frac{f(c)}{3}$ for some $c \in [0,1]$
154881
Question on differentiability of a continuous function