Suppose $f$ is differentiable on $R$ and $f'(x) \leq 4$ for all $x ∈ R$. Prove that there is at most one $a \leq 2$ such that $f(a) = a^2$.
101570
If derivative of function is less than or equal to 4, prove the existence of point $c$ so $f(c)=c^2$.
15032
Let $f: \Bbb R \to \Bbb R$ be a differentiable function such that $\sup_{x \in \Bbb R}|f'(x)| \lt \infty$. Then
98462
Prove the inverse of a strictly increasing function is differentiable.
60348
How do I prove $F(a)=F(a^2)?$
111458
Prove that $f$ is differentiable
30204
Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
20192
Prove that: $(f(x))^{2} + (f'(x))^{2} \leqslant \max(a,b)$ where $(a,b) \in \mathbb{R}^2$
30855
Finding all differentiable $f(z) = u(x) + iv(y)$ in $\mathbb{C}$ where $u(x),v(y)$ are real valued functions.
33130
$f$ differentiable except in $1$ point
91430
If $f$ is twice differentiable, $\big(f(y) - f(x)\big)/(y-x)$ is differentiable
96450
If $f(x)$ is continuous and differentiable function , prove that $f(x) \geq e^{3x}, \forall x \geq 0$
48592
Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
2717
Help proving existence of an specific $x_0$ for which $f''(x_0)=0$
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
112923
Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ a differentiable function such that $f'(x)=0$ for all $x\in\mathbb{Q}$
145173
$f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$
111470
Proof of differentiability of $f(x)$