$|f''(x)|\leq M$ implies $|f(c)-f(a)-(c-a)\frac{f(b)-f(a)}{b-a}|\leq \frac{1}{8} M(b-a)^2$
39451
Proving inequation with maximum and 2nd derivative
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
5369
Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
15441
Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$
51347
Prove that $|a|\leq \max\{|b|,|c|\}$ if $b\leq a \leq c$
73987
Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$
112565
Prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\geq \frac{9}{a+b+c} : (a, b, c) > 0$
4065
What are the entire functions $f$ such that $|f'(z)| \leq |f(z)| \ ? $
139453
Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
15116
Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
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
4498
Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
127759
Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$
95288
Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$.