Prove $f(x)\geq e^x$
96450
If $f(x)$ is continuous and differentiable function , prove that $f(x) \geq e^{3x}, \forall x \geq 0$
45742
Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$
90896
proof of inequality $e^x\le x+e^{x^2}$
128790
Prove an integral inequality: $ \left(\int|f|^2dx\right)^2\le 4\left(\int|xf(x)|^2dx\right)\left(\int|f'|^2dx\right) $
59513
For $a, b \geq 0$, $0 < x < 1$, show $(a+b)^x \leq a^x + b^x$
79281
Let $f : X \to Y$ be a function and $E \subseteq X$ and $F \subseteq X$. Show that in general
111186
Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
116056
Prove that $|f(x)-f(y)|\le|x-y|$ for all $x$ and $y$.
61799
Prove that $\Vert f(b)-f(a)-f'(a)(b-a) \Vert\leq \sup_{x\in [a,b]} \Vert f''(x)\Vert\Vert b-a\Vert^2$
69656
Prove or disprove that $\int_a^b |f(x)|\ \mathrm{d}x\geq \big | \int_a^b f(x)\ \mathrm{d}x\big| $
8179
If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
142922
How to prove that $e^x$ is convex?
146180
Prove that $x\sqrt{1-x^2} \leq \sin x \leq x$
117680
Prove that $|e^{i\theta_1}-e^{i\theta_2}| \leq |\theta_1 - \theta_2|$
20192
Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
110368
Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$