If $f(0)=0 ,f(1)=1 ,f'(0)=f'(1)=0$, then $|f''(x)|>4$
57026
$f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
13927
Values of $a$ s.t. for all continuous $f$ with $f(0)=f(1)$ there exists $x$ s.t. $f(x+a) = f(x)$
57193
Prove that $f$ has derivatives of all orders at $x=0$, and that $f^{(n)}(0)=0$ for $n=1,2,3\ldots$
32525
If $f(1)=1$, then is it true that $f(n)=n$ for all $n \in \mathbb{N}\cup\{0\}$.
21954
A question about $f(x)\equiv 0$
95970
Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
17154
Does a non-trivial solution exist for $f'(x)=f(f(x))$?
35556
$f(z)=\frac{z}{1+|z|} $ Is 1-1?
48592
Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
129579
If $f''(x_0)$ exists then $\lim_{x \to x_0} \frac{f(x_0+h)-2f(x_0)+f(x_0-h)}{h^2} = f''(x_0)$
133682
Prove that if $f(t)$ satisfies $f(t)=f'(t)$, then $f(t+s)=f(t)f(s)$ for all $t,s\in R$ without using the fact that $f(t)=c\cdot e^t$.
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).
145424
If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
106013
Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable
104263
How to show that $f'(x)<2f(x)$
13202
Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$