Chapter 5.6, Problem 93E

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

Extended Mean Value Theorem In Exercises 91-94, verify that the Extended Mean Value Theorem can be applied to the functions f and g on the closed interval [a, b]. Then find all values c in the open interval (a, b) such that f ' ( c ) g ' ( c ) = f ( b ) − f ( a ) g ( b ) − g ( a ) . FunctionInterval f ( x ) = sin   x ,     g ( x ) = cos   x [ 0 , π 2 ]

To determine

To calculate: The value of c in the open interval (0,π2) of function f(x)=sinx,g(x)=cosx and verify the Extended Mean Value theorem can be applied to the functions f and g on the closed interval [0,π2].

Explanation

Given:

The functions f and g are,

f(x)=sinx,g(x)=cosx

And the closed interval [0,Ï€2].

Formula used:

Differentiate the function and use the Extended Mean Value Theorem,

fâ€²(x)gâ€²(x)=f(b)âˆ’f(a)g(b)âˆ’g(a)

Calculation:

The functions f and g are,

f(x)=sinx,g(x)=cosx

And the closed interval [0,Ï€2] then the values of functions are,

f(0)=0,g(0)=1, f(Ï€2)=1,g(Ï€2)=0 and fâ€²(c)=cosc,gâ€²(c)=âˆ’sinc

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