Prove that if $f'(x)≠0$ on an interval $( a , b)$ on $A$ for a differentiable function $f$, then $f$ is one-to-one on $A$.
13150
If the derivative of $f$ is never zero, then $f$ is one-to-one
49408
Proof of the derivative of ln(x)
143491
Show that $f$ is differentiable on $(−1, 1)$.
97104
Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
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).
11167
A counterexample of a differentiable function on an interval whose derivative is discontinuous
104263
How to show that $f'(x)<2f(x)$
76908
If $f'$ is differentiable at $a$ then $f'$ is continuous at $(a-\delta,a+\delta)$
111485
Let $f(x)$ be continuous on [$0,1$] to $f(x)=0$ for each $x\in[0,1]$
96450
If $f(x)$ is continuous and differentiable function , prove that $f(x) \geq e^{3x}, \forall x \geq 0$
74583
Prove that $f'(0)=L$.
112285
Show that $\lim_{x\to\infty} f(x)=0 \implies \lim_{x\to\infty} f'(x)=0$
53497
Prove that the function is continuous and differentiable (not as easy as it sounds imo)
145424
If $f''(x)+f(x)>0$ and $f(x)>0$ $\forall x\in(a,b)$; $f(a)=f(b)=0$; prove that $b-a>\pi$.
30860
Prove that $f(x)=0$ in $[-1,1]$.
131793
Zero integral of measurable $f$ on every interval implies $f=0$?
130322
Show that $f(x) = 0$ if $x = 0$ $f(x) = \text{exp}(-1/x^{2})$ if $x \neq 0$ is infinitely differentiable but not analytic at $x_{0} = 0$ `