If $f(x),g(x)$ are two differentiable functions such that $f'(x)=g(x)$, $|f(x)|<1$ and $f(0)^2+g(0)^2=9$
122330
Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.
18905
$f(a)=f(b)=f_{+}'(a)=f_{-}'(b)=0$ and $|f''(x)|\le M$. Show that $|f(x)|\le \frac{M(b-a)^2}{16}$.
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
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
20273
Prove that if $|f(x)| \leq x^2$, then the function is continuous and differentiable at $x=0$.
26424
If $f(x)<g(x)$ prove that $\lim f(x)<\lim g(x)$
63750
let $f(\frac{x}{3})+f(\frac{2}{x})=\frac{4}{x^2}+\frac{x^2}{9}-2$ then find $f(x)$
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
105319
A calculus problem with functions such that $f''(x) = g(x)$ and $g''(x) = f(x)$
145424
If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
24810
Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
47688
Number of roots of differentiable function
62270
Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$
26207
If $f$ is a quadratic and $f(x)>0\;\forall x$, and $g= f + f' + f''$, prove $g(x)>0\; \forall x$
28315
Monotone functions and non-vanishing of derivative
56413
$f(g(x))=g(f(x))$ implies $f(c)=g(c)$ for some $c$
150005
Existence of points in a differentiable function