   Chapter 3.2, Problem 28E

Chapter
Section
Textbook Problem

# Suppose that f and g are continuous on [a, b] and differentiable on (a, b). Suppose also that f ( a ) = g ( a ) and f ′ ( x ) < g ′ ( x ) for a   <   x   <   b . Prove that f ( b )   <   g ( b ) . [Hint: Apply the Mean Value Theorem to the function h = f − g . ]

To determine

To prove:

f(b)<g(b)

Explanation

1) Concept:

Use the Mean Value Theorem to verify.

2) Theorem:

Mean value theorem: If f is a function that satisfies the following hypotheses:

1) f is continuous on [a,b]

2) f is differentiable on (a,b)

Then, there is number c in (a,b) such that f'c=fb-f(a)b-a

3) Given:

f  and g are continuous on [a,b] and differentiable on (a,b)

fa=g(a)

f'(x)<g'(x) for a<x<b

4) Calculation:

Let, h(x)=f(x)-g(x)

f and g are continuous on [a,b] hence h is so

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