$f(0)=g(0)$ and $f'(x) \leq g'(x) ~\forall~ x \in \mathbb{R} \implies f(x) \leq g(x) ~\forall~ x>0$
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
138544
For which values of $x$, will $f (x) . f '(x) < 0$?
96450
If $f(x)$ is continuous and differentiable function , prove that $f(x) \geq e^{3x}, \forall x \geq 0$
35889
Suppose that f(0) = f(1) = 0 and x ∈ (0, 1). Prove that there exists c ∈ (0, 1) such that f'(c) = f(x).
42853
If $f \geq 0$ is continuous and $\int_{a}^{b} f(x) \, dx = 0$, then $f =0$
48275
$f$ defined $|f(x) - f(y)| \leq |x - y|^{1+ \alpha}$ Prove that $f$ is a constant.
32506
Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
149739
Prove: $f(t)\leq t+1$
13202
Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$
64955
Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
37308
Let $f:K\to K$ with $\|f(x)-f(y)\|\geq ||x-y||$ for all $x,y$. Show that equality holds and that $f$ is surjective.
33523
If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing.
111186
Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
95626
$\int_X |f_n - f| \,dm \leq \frac{1}{n^2}$ for all $n \geq 1$ $\implies$ $f_n \rightarrow f$ a.e.
7825
If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$
4656
Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions