Proving Inequalities Using MVT

Real Analysis · Axiom Academy

EXAMPLE Proving Inequalities Using the MVT Use the Mean Value Theorem to prove that for every . Prove that for all . The strategy: build the auxiliary function , apply the Mean Value Theorem on , and bound the derivative f'(c) to force . For every the line y = x stays strictly above the curve . That vertical gap is exactly , and the proof shows it is always positive. Nice work. You proved a clean inequality by turning it into a statement about a derivative and bounding that derivative with the MVT. Pick the right auxiliary function: to prove , set so that is exactly the inequality you want. MVT turns a gap into a derivative: on it gives a c with . Bound the derivative: everywhere, and it is positive except at isolated points, so f is strictly increasing. Result: since f(0) = 0 and f increases, for , which is , i.e. . The same recipe — auxiliary function, MVT, bound the derivative — proves a huge family of inequalities such as and .

This is the written version of the interactive lesson above. See the full Real Analysis course.