Let a be any number in I

Then the tangent line at x = a is y(x) = f ' (a)(x-a) + f(a)

Let h(x) = f(x) - y(x) x>a

h '(x) = f '(x) - f ' (a) > 0 for x > a since if f '' (x) is positive then f ' (x) is increasing

Therefore h(x) is an increasing function

h(a) = f(a) - y(a) = 0

Therefore h(x) > h(a) > 0 i.e f(x) > y(x) for all x > a

Since this is true for any a in I the result follows