A question in my math book is: "Use mathematical induction to show that:

$$e^x>1+x+\frac{x^2}{2!}+...+\frac{x^n}{n!}$$

if $x>0$ and $n$ is any positive integer."

Apparently the solution involves applying the Mean Value Theorem on:

$$g(t)=e^t-1-t-\frac{t^2}{2!}-...-\frac{t^{k+1}}{(k+1)!}$$

Then: 

$$\frac{g(x)}{x}=\frac{g(x)-g(0)}{x-0}=g'(c)$$

Then it states that $g'(c)$ is positive.

QUESTION 1: why is $g'(c)$ positive?

QUESTION 2: how would one answer (in small understandable steps) the original question of my math book? I don't fully understand the method used.

Thanks!