Suppose $f'(x_0)$ exists and is positive, then there exists $x_1 > x_0$ such that $f(x) > f(x_0)$ for all $x \in (x_0,x_1)$.
What I have done so far: Since $f(x_0) > 0$ and exists, then $\lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}=f'(x_0) > 0$. Then let $x_1 > x_0$. So $f'(x_0)(x_1-x_0)=x_1*f'(x_0) - x_0*f'(x_0) = x_1 * \lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0} - x_0 * \lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}$
I'm not sure whether I am going about this in the right direction and whether I can just deal with the inside part of the limit. Let me know if I'm doing this right so far. Thanks in advance.