Mean Value Theorem Math Example 2

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Example 2

hard
Use the MVT to prove: if f′(x)=0f'(x) = 0 for all xx in (a,b)(a, b), then ff is constant on (a,b)(a, b).

Solution

  1. 1
    Let x1,x2∈(a,b)x_1, x_2 \in (a,b) with x1<x2x_1 < x_2.
  2. 2
    ff is continuous on [x1,x2][x_1, x_2] and differentiable on (x1,x2)(x_1, x_2) (given).
  3. 3
    By MVT: f(x2)−f(x1)=f′(c)(x2−x1)f(x_2) - f(x_1) = f'(c)(x_2 - x_1) for some c∈(x1,x2)c \in (x_1, x_2).
  4. 4
    Since f′(c)=0f'(c) = 0: f(x2)−f(x1)=0f(x_2) - f(x_1) = 0, so f(x2)=f(x1)f(x_2) = f(x_1).
  5. 5
    Since x1,x2x_1, x_2 were arbitrary, ff is constant on (a,b)(a,b).

Answer

f′(x)=0f'(x) = 0 on (a,b)⇒f(a,b) \Rightarrow f is constant on (a,b)(a,b).
This is a fundamental corollary of the MVT. The proof works by picking any two points and showing the MVT forces equal function values. It also implies two antiderivatives of the same function differ by a constant.

About Mean Value Theorem

If ff is continuous on [a,b][a, b] and differentiable on (a,b)(a, b), then there exists at least one point cc in (a,b)(a, b) where f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}

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More Mean Value Theorem Examples

Example 1 easy

Verify the MVT for [formula] on [formula] by finding the value of [formula].

Example 3 easy

Find the value of [formula] guaranteed by the MVT for [formula] on [formula].

Example 4 medium

A car travels 120 miles in 2 hours. Explain why the MVT guarantees the car exceeded 60 mph at some i

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