Show that $|f''(\mu)|\geq\frac{4}{(b-a)^2}[f(b)-f(a)]$ for some $\mu\in(a,b)$ provided that $f'(a)=f'(b)=0$
34081
Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$
257
Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$
89429
Prove $\sup \left| f'\left( x\right) \right| ^{2}\leqslant 4\sup \left| f\left( x\right) \right| \sup \left| f''\left( x\right) \right| $
2414
If $f \in \operatorname{Hol}(D)$, $f(\frac{1}{2}) + f(-\frac{1}{2}) = 0$, prove that $|f(0)| \leq \frac{1}{4}$
20192
Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
149764
If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$
101700
Liouvilles theorem question - show $f$ is constant
17646
If $\frac{f(x)}{x}$ is decreasing in $(0,\infty)$, prove that $f(a+b) \leq f(a) + f(b)$ for $a,b >0$
150699
Show that if $f'$ is strictly increasing, then $\frac{f(x)}{x}$ is increasing over $(0,\infty)$
18261
Prove $\int_{a}^{b}{f(x)dx}>0$.
127759
Prove that $\frac{\sin(a)}{\sin(b)} < \frac{a}{b} < \frac{\tan(a)}{\tan(b)}$ where $0 < b < a < \frac{\pi}{2}$
8854
Prove $f $ is identically zero if $f(0)=0$ and $|f'(x)|\le|f(x)|$
13202
Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$
128495
Integral Inequality $|f''(x)/f(x)|$
91877
Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
95288
Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$.
97643
Prove that:$f(f(x)) = x^2 \implies \int_{0}^{1}{(f(x))^2dx} \geq \frac{3}{13}$