It is a typical application of the Mean Value Theorem to a convex function (i.e. a function whose derivative f' is an increasing function).
Mean Value Theorem: if f is continuous and its derivative exists in the interval (a,b), then
f(b)-f(a) = f'(c)(b-a)
for some c in (a,b).
If f' is an increasing function, then a < c < b implies f'(a) < f'(c) < f'(b), but now (thanks to the Mean Value Theorem) you have an expression for f'(c). The last step is choosing the right convex function for your inequality. In fact, you can use the Mean Value Theorem to obtain many inequalities from many functions.