Prove the following equality involving $f'(0)$
139453
Prove that $\frac{1}{f'(a)} = \frac{1}{2\pi i} \int_C \frac{1}{f(z)-f(a)} dz$
64339
Find $f$ where $f'(x) = f(1+x)$
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
110368
Prove the following integral inequality: $\int_{0}^{1}f(g(x))dx\le\int_{0}^{1}f(x)dx+\int_{0}^{1}g(x)dx$
23864
Rolle's theorem $\beta \cdot f(x)+f'(x)=0$
136980
Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
58832
How prove this $\int_{a}^{b}[f''(x)]^2dx\ge\dfrac{4}{b-a}$
103706
If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
158180
An inequality of $\int_0^1 |f(x)|dx$
111186
Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$
16418
Showing that $\frac{f(x)-f(x-h)}{h} \le f'(x) \le \frac{f(x+h)-f(x)}{h}$ given that f' is increasing
13270
Prove $\exists \xi \in (a,b):f'(\xi)<1+f^2(\xi)$
86409
Show that $\lim_{x \to +\infty}\left(f(x)+f'(x)\right)=0 \Rightarrow \lim_{x \to +\infty} f(x)=0$
17154
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
95288
Prove that, $f'(0) \ge -\sqrt{2}$ for a function $f$ satisfying some conditions on $(-1,1)$.
58943
If $f'(x)\le g'(x)$, prove $f(x)\le g(x)$
75653
Prove that $|f ''|\ge 4$