f'(x)>0 --> f inc Theorem

If $f^{\prime }(x)>0$ then $f(x)$ is Strictly Increasing.

Theorem

Let $f$ be a diff function on an open interval $I$.

1. if $x\in I,f^{\prime }(x)>0$ then $f(x)$ is increasing on $I$
2. If $x\in I,f^{\prime }(x)<0$ then $f(x)$ is decreasing on $I$

Proof (1)

1. Assume $f^{\prime }(x)>0$
2. Then $f(x)$ is diff on $I$
3. Then $f(x)$ is cont on $I$ as Differentiable Implies Continuity Proof
4. Let $x_1,x_2\in I$ be arbitrary
5. Assume $x_1<x_2$
6. Note that $f$ is cont on $[x_1,x_2]$
7. Note that $f$ is diff on $(x_1,x_2)$
8. Then consider $f^{\prime }(x)=\frac{f(x_2)-f(x_1)}{x_2-x_1}$
   1. $⟹0<\frac{f(x_2)-f(x_1)}{x_2-x_1}$
   2. $⟹0<f(x_2)-f(x_1)$
   3. $⟹f(x_1)<f(x_2)$ As required $\square$