Chapter 4.2, Problem 55E

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

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 log , x ,     [ 1 , 2 ]

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)=xlog2x on the closed interval [1,2] if cannot apply mean value theorem than explain it.

Explanation

Given:

The provided function f(x)=xlog2x on the closed interval [1,2].

And,

fâ€²(c)=f(b)âˆ’f(a)bâˆ’a

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,

fâ€²(c)=f(b)âˆ’f(a)bâˆ’a

Derivative formula:

ddxlnx=1x

The logarithm property:

logaa=1

And,

ddxuv=uvâ€²+vuâ€²

Calculation:

Consider the function:

f(x)=xlog2x

Use the log property logax=lnxlna.

Thus, the above function f is written as;

f(x)=xlog2x=xlnxln2

This function f is continuous on [1,2] and differentiable on (1,2)

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