Chapter 4, Problem 11PS

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Proof Let f and g be functions that are continuous on [a, b] and differentiable on (a, b). Prove that if f ( a ) = g ( a ) and g ' ( x ) > f ' ( x ) for all x in (a, b), then g ( b ) > f ( b ) .

To determine

To prove: If for two functions f and g, f'(x)<g'(x) for all a<x<b and f(a)=g(a), then g(b)>f(b).

Explanation

Given:

The functions f and g are continuous on the interval [a,b] and differentiable on the interval (a,b).

Formula used:

The Mean Value Theorem states that if a function g(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) then for any câˆˆ(a,b),

g'(c)=g(b)âˆ’g(a)bâˆ’a

Proof:

Define the following function:

h(x)=g(x)âˆ’f(x)

Clearly, the function h is continuous on the interval [a,b] and differentiable on the interval (a,b) due to the properties of functions f and g

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