$f:[0,1] \to \mathbb{R}$ with $f(0) = 0 = f(1)$ and $|f''|\leq M$. Show $|f'(1/2)| \leq \frac{M}{4}$.
111186
Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
17646
If $\frac{f(x)}{x}$ is decreasing in $(0,\infty)$, prove that $f(a+b) \leq f(a) + f(b)$ for $a,b >0$
51659
If $f(0)=0$ then prove that $|f(z)+f(-z)| \leq 2|z|^2$
134195
Holomorphic function $f:B(0,1) \to B(1,4)$ with $f(0)=3$ and $f(\frac{1}{2})=1$
64955
Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
30860
Prove that $f(x)=0$ in $[-1,1]$.
26207
If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
104263
How to show that $f'(x)<2f(x)$
59063
$f(f(x))=f(x)$ question
121771
Proof $f(x)\equiv 0$
76580
If $f(\frac{1}{n})=0$ all $ n\to \mathbb{Z^{+}}$ then find $ f'(0)$?
85739
Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
139453
Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
69707
If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
71225
$f(0)=0$ and $\lim_{x \to \infty}f(x)=0$. Show there's $x_0$ such that $f'(x_0)=0$
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
28125
Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$