Lef f be a real-valued function defined on an open interval I. Suppose that x_0 is in I and that f is differentiable at x_0
Prove that if f'(x_0) > 0 then there is a positive number delta such that for every x in I with x_0 < x < x_0 + delta we have f(x) > f(x_0), and for every x in I with x_0 - delta < x < x_0 we have f(x) < f(x_0).
Prove that if f'(x_0) > 0 then there is a positive number delta such that for every x in I with x_0 < x < x_0 + delta we have f(x) > f(x_0), and for every x in I with x_0 - delta < x < x_0 we have f(x) < f(x_0).