Let $f(x)$ be differentiable over $\left[0,1\right]$ with $f(0) = f(1) = 0$. Prove that $f'(x) - 2f(x)$ has a zero in $(0,1)$
32506
Prove $kf(x)+f'(x)=0 $ when conditions of Rolle's theorem are satisfied .
104510
For continuous $f$, $\int_0^x f(t)dt = ∫_x^1f(t)dt$ implies $f=0$.
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
98344
Prove that $\int_0^1[f''(x)]^2dx\ge4.$
101778
Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
113888
Behavior of derivative near the zero of a function.
69008
$f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
78834
if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
67020
Proving that for a smooth function if $f(\frac 1 k)=0 :\forall k\in \mathbb N$ then $f(x)=0 :\forall x\in[-1,1]$
75390
How does one prove that if function's partial derivative respect to every variable is zero, function is constant?
143491
Show that $f$ is differentiable on $(−1, 1)$.
61782
How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
61190
If all derivatives of $f$ are uniformly bounded by a common constant and $f(1/n) = 0$ for all $n$, is $f$ identically zero?
121771
Proof $f(x)\equiv 0$
136980
Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
105084
Let f be a function twice differentiable and with derivatives continuous on an interval $[a,b]$ containing $0$. Prove the following statement:
148737
Is $|xy|$ differentiable in $(0,0)$?