How prove this $f(0)=f(1)$,
49099
Constructing a continuous function whose graph seems 'special'
107727
$m\{x\in [0,1]:f'(x)=0\}>0$
82001
How to prove the function is a constant
57193
Prove that $f$ has derivatives of all orders at $x=0$, and that $f^{(n)}(0)=0$ for $n=1,2,3\ldots$
118873
Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
95469
Suppose $f:[0,1] \Rightarrow \mathbb{R}$ is continuous and $\int_0^x f(x)dx = \int_x^1 f(x)dx$. Prove that $f(x) = 0$ for all $x$
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
1057
How prove this integral limit $=f(\frac{1}{2})$
85739
Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|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 $b-a>\pi$.
132013
How do I show that if $f : [0,1] \to [0,1]$ is continuous, then there is some $c$ such that $f(c) = c$?
57026
$f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
69008
$f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
101778
Show that there is $\eta\in(0,1)$ with $f''(\eta)=f(\eta)$
131692
How to show that $f_{0}f_{1}+f_{1}f_{2}+...+f_{2n-1}f_{2n} = f^{2}_{2n}$
120774
Show that $f$ has exactly one zero on the square $Q =$ {$x + iy ∈ \Bbb C : |x| < 1, |y| < 1$}.
111458
Prove that $f$ is differentiable