Prove that for a polynomial function $f$ if $f(x) \geq 0$ for all $x$, then $f(x) + f'(x) + \cdots + f^{(n)} (x) \geq 0$
83733
sum of polynoms of given property
133682
Prove that if $f(t)$ satisfies $f(t)=f'(t)$, then $f(t+s)=f(t)f(s)$ for all $t,s\in R$ without using the fact that $f(t)=c\cdot e^t$.
4498
Proof that $\|fx\| \leq \|f\|\cdot\|x\|$
82567
Prove that $ f(x) $ has at least two real roots in $ (0,\pi) $
111756
How do I prove that then $f(x) = 1$ for some $x ∈ [0, 1]$?
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 $x ≤ 0$.
127056
How to prove that a polynomial of degree $n$ has at most $n$ roots?
28007
Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $
95293
Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
57368
Prove that a positive polynomial function can be written as the squares of two polynomial functions
26219
Suppose f is a function such that f'(x) = 1/x for all x > 0. Prove that if f(1) = 0, then f(xy) = f(x) + f(y) for all x,y > 0
118873
Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
82591
Prove $(x+r_1) \cdots (x+r_n) \geq (x+(r_1 \cdots r_n)^{1/n})^{n}$.
48592
Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
27570
Prove $(1-x)\ln(1-x)+(1+x)\ln(1+x)\leq 2x^2$ for $0<x<1$
51653
Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$
85739
Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $