Let $f$ be a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)f''(x)\leq 0$ for all $x\in \mathbb{R}$.
115712
Application of Derivatives rigorous proof
64955
Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
158561
Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is
135577
Prove the existence of $c$ such that $f'(c) = 2c(f(c) - f(0))$
75638
Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq f(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
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
63526
Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
6636
Is there a positive function $f$ on real line such that $f(x)f(y)\le|x-y|, \forall x\in \mathbb Q , \forall y \in \mathbb R \setminus \mathbb Q$?
69869
if $f$ is continuous on $\mathbb{R}$ and $f(r)=0,r \in \mathbb{Q}$, then $f(x)=0,x \in \mathbb{R}$
88311
Linear functional f(v)=0 imply v=0
90565
If $f:[0,1] \rightarrow \mathbb{R}$ is continuous function such that $f(0)=f(1)$ then there exists $ x\in [0,1]$ such that $f(x) = f(x+\frac{1}{n})$.
105172
Prove that the entire function is constant over certain conditions.
45342
Generalization of Mean Value Theorem to functions, $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$
69695
$f\geq 0$, continuous and $\int_a^b f=0$ implies $f=0$ everywhere on $[a,b]$
81758
Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero.
21954
A question about $f(x)\equiv 0$