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MathidiaMathematics Academy
Further calculus
Lesson 45 of 75
Review

Calculus proof

Objective. Use monotonicity, convexity and integration in rigorous arguments.

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Learn: the key idea

Calculus proof uses properties such as monotonicity (where a function is strictly increasing or decreasing, determined by the sign of its derivative), convexity (determined by the sign of the second derivative), and integration to construct rigorous mathematical arguments rather than numerical checks. For instance, to prove an inequality like eˣ ≥ 1+x for all real x, one can show f(x)=eˣ−1−x has f(0)=0, f'(x)=eˣ−1 which is negative for x<0 and positive for x>0, so f has a minimum at x=0 where f(0)=0, proving f(x)≥0 everywhere. This technique of analysing a difference function's derivative is a standard tool for proving inequalities.

Worked example

Prove that ln x ≤ x − 1 for all x > 0, using properties of the derivative of a suitable function.

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Illustrated card showing 1 … 0
  1. 1Let f(x) = x − 1 − ln x for x > 0
  2. 2Differentiate: f'(x) = 1 − 1/x, which is zero at x=1, negative for 0<x<1, positive for x>1
  3. 3So f has a minimum at x=1, where f(1) = 1 − 1 − ln 1 = 0
  4. 4Since the minimum value of f is 0, f(x) ≥ 0 for all x>0, so ln x ≤ x − 1

Answer: Proved: ln x ≤ x − 1 for all x > 0

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Common mistake. Students 'prove' an inequality by checking a few numerical values rather than analysing the sign of a derivative over the whole domain. A proof must hold for every value in the stated range, which requires calculus reasoning, not spot-checks.

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