a. Answer The local maximum at ( 8 3 , 16 3 3 ) The local minimum at ( 4 , 0 ) Explanation Given information : The given function is f ( x ) = x 4 − x Concept used : Extreme values occur only at critical points and end points. A point in the interior of the domain of a function f at which f ′ = 0 or f ′ does not exist is called a critical point of f . Calculation : The domain of the function is ( − ∞ , 4 ] Take derivative of the function f ′ ( x ) = − x 2 4 − x + 4 − x Equate the derivative to 0. − x 2 4 − x + 4 − x = 0 x 2 4 − x = 4 − x x = 2 ( 4 − x ) x = 8 − 2 x 3 x = 8 x = 8 3 Value at x = 8 3 f ( x ) = 8 3 4 − 8 3 f ( x ) = 8 3 4 3 f ( x ) = 16 3 3 x = 4 Is a critical point and the value at x = 4 f ( x ) = 4 4 − 4 f ( x ) = 0 Therefore, The local maximum at ( 8 3 , 16 3 3 ) The local minimum at ( 4 , 0 ) b. To determine To find the intervals on which the function is increasing. Answer The function increases on ( − ∞ , 8 3 ] . Explanation Given information : The given function is f ( x ) = x 4 − x Concept used : If f ′ > 0 at each point of ( a , b ) , then f increases on [ a , b ] It is assumed that f be continuous on [ a , b ] and differentiable on ( a , b ) . Calculation : For increasing function f ′ ( x ) must be greater than 0. − x 2 4 − x + 4 − x > 0 − x 2 4 − x > − 4 − x x 2 4 − x < 4 − x x < 2 ( 4 − x ) x < 8 − 2 x 3 x < 8 x < 8 3 Therefore, The function increases on ( − ∞ , 8 3 ] . c. To determine To find the intervals on which the function is decreasing. Answer The function decreases on [ 8 3 , 4 ] Explanation Given information: The given function is f ( x ) = x 4 − x Concept used : If f ′ < 0 at each point of ( a , b ) , then f decreases on [ a , b ] It is assumed that f be continuous on [ a , b ] and differentiable on ( a , b ) . Calculation : For decreasing function f ′ ( x ) must be less than 0. − x 2 4 − x + 4 − x < 0 − x 2 4 − x < − 4 − x x 2 4 − x > 4 − x x > 2 ( 4 − x ) x > 8 − 2 x 3 x > 8 x > 8 3 Therefore, The function decreases on [ 8 3 , 4 ]

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Videos

Question

Chapter 5.2, Problem 23E

To determine

To find the local extrema.

Expert Solution

Answer to Problem 23E

The local maximum at (83,1633)

The local minimum at (4,0)

Explanation of Solution

Given information:

The given function is f(x)=x4−x

Concept used:

Extreme values occur only at critical points and end points.

A point in the interior of the domain of a function f at which f′=0 or f′ does not exist is called a critical point of f .

Calculation :

The domain of the function is (−∞,4]

Take derivative of the function

f′(x)=−x24−x+4−x

Equate the derivative to 0.

−x24−x+4−x=0x24−x=4−xx=2(4−x)x=8−2x3x=8x=83

Value at x=83

f(x)=834−83f(x)=8343f(x)=1633

x=4 Is a critical point and the value at x=4

f(x)=44−4f(x)=0

Therefore,

The local maximum at (83,1633)

The local minimum at (4,0)

To determine

To find the intervals on which the function is increasing.

Expert Solution

Answer to Problem 23E

The function increases on (−∞,83] .

Explanation of Solution

Given information:

The given function is f(x)=x4−x

Concept used:

If f′>0 at each point of (a,b) , then f increases on [a,b]

It is assumed that f be continuous on [a,b] and differentiable on (a,b) .

Calculation :

For increasing function f′(x) must be greater than 0.

−x24−x+4−x>0−x24−x>−4−xx24−x<4−xx<2(4−x)x<8−2x3x<8x<83

Therefore,

The function increases on (−∞,83] .

To determine

To find the intervals on which the function is decreasing.

Expert Solution

Answer to Problem 23E

The function decreases on [83,4]

Explanation of Solution

Given information:

The given function is f(x)=x4−x

Concept used:

If f′<0 at each point of (a,b) , then f decreases on [a,b]

It is assumed that f be continuous on [a,b] and differentiable on (a,b) .

Calculation :

For decreasing function f′(x) must be less than 0.

−x24−x+4−x<0−x24−x<−4−xx24−x>4−xx>2(4−x)x>8−2x3x>8x>83

Therefore,

The function decreases on [83,4]

Chapter 5.2, Problem 22E

Chapter 5.2, Problem 24E

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 1QR Ch. 5.1 - Prob. 2QR Ch. 5.1 - Prob. 3QR Ch. 5.1 - Prob. 4QR Ch. 5.1 - Prob. 5QR Ch. 5.1 - Prob. 6QR Ch. 5.1 - Prob. 7QR Ch. 5.1 - Prob. 8QR Ch. 5.1 - Prob. 9QR Ch. 5.1 - Prob. 10QR

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