Using Derivative for Comparing Functions: Survey of Methods

If you wish to simultaneously follow another text on this topic in a separate window, click here for Theory and here for Solved Problems.

Let f and g be functions on an interval I from a to b.

Proving that f = g on I:
Step 1. Prove that f ′ = g′ on Int( I ).
Step 2. Find some c from I such that f (c) = g(c), or show that f (a⁺) = g(a⁺) or show that f (b^-) = g(b^-) (recall that this notation denotes one-sided limits).

Proving that f  is constant on I:
Step 1. Prove that f ′ = 0 on Int( I ).
Step 2. The value of the constant can be obtained as f (c) for some c from I, or as f (a⁺) or f (b^-).

Proving that f  ≤ g on I:
Step 1. Prove that f ′ ≤ g′ on Int( I ).
Step 2. Show that f (a) ≤ g(a) or that f (a⁺) ≤ g(a⁺).
Alternative:
Step 1. Prove that f ′ ≥ g′ on Int( I ).
Step 2. Show that f (b) ≤ g(b) or that f (b^-) ≤ g(b^-).

Proving that f  ≥ 0 on I:
Step 1. Prove that f ′ ≥ 0 on Int( I ).
Step 2. Show that f (a) ≥ 0.

Versions for sharp inequalities:

Proving that f  < g on I:
Step 1. Prove that f ′ < g′ on Int( I ).
Step 2. Show that f (a) ≤ g(a) or that f (a⁺) ≤ g(a⁺).

Proving that f  > 0 on I:
Step 1. Prove that f ′ > 0 on Int( I ).
Step 2. Show that f (a) ≥ 0.

For some examples see the section Tricks with MVT in Theory - MVT, or Solved Problems.