   Chapter 4.2, Problem 54E Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

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

Using the Mean Value Theorem In Exercises 43-56, determine whether the Mean Value Theorem can be applied to f on the closed interval [ a, b ]. If the Mean Value Theorem can be applied, find all values of c in the open interval ( a,b ) such that f ' ( c ) = f ( b ) − f ( a ) b − a . If the Mean Value Theorem cannot be applied, explain why not. f ( x ) = ( x + 3 ) ln ( x + 3 ) ,     [ − 2 , − 1 ]

To determine

To calculate: The all values of c in the open interval (a,b) and also find that the mean value theorem can be applied to f(x)=(x+3)ln(x+3) on the closed interval [2,1] if cannot apply mean value theorem than explain it.

Explanation

Given:

The provided function f(x)=(x+3)ln(x+3) on the closed interval [2,1].

And,

f(c)=f(b)f(a)ba

Fomula used:

The Mean Value Theorem state that,

The function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) so, there exists at least one number c in (a,b) such that,

f(c)=f(b)f(a)ba

Derivative formula:

ddxlnx=1x

And,

ddxuv=uv+vu

Calculation:

Consider the function:

f(x)=(x+3)ln(x+3)

This is logarithmic function, which is always continuous and differentiable for all positive and negative real values in the open interval (3,3)

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