Calculus: Graphical, Numerical, Algebraic

3rd Edition

ISBN: 9780133688399

Author: Finney, Ross L.

Publisher: Pearson/Prentice Hall

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 5.2, Problem 16E

To determine

To find the local extrema.

Expert Solution

Answer to Problem 16E

The local minimum at (12,−494)

Explanation of Solution

Given information:

The given function is g(x)=x2−x−12

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 :

Take derivative of the function

g′(x)=2x−1

Equate the derivative to 0.

2x−1=02x=1x=12

Put the value of x in the function

g(12)=(12)2−12−12g(12)=14−12−12g(12)=−494

Therefore,

The local minimum at (12,−494)

To determine

To find the intervals on which the function is increasing.

Expert Solution

Answer to Problem 16E

The function increases in the interval [12,∞)

Explanation of Solution

Given information:

The given function is g(x)=x2−x−12

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 g′(x) must be greater than 0.

2x−1>02x>1x>12

Therefore,

The function increases in the interval [12,∞)

To determine

To find the intervals on which the function is decreasing.

Expert Solution

Answer to Problem 16E

The function decreases in the interval (−∞,12]

Explanation of Solution

Given information:

The given function is g(x)=x2−x−12

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 g′(x) must be less than 0.

2x−1<02x<1x<12

Therefore,

The function decreases in the interval (−∞,12]

Chapter 5.2, Problem 15E

Chapter 5.2, Problem 17E

Chapter 5 Solutions

Calculus: Graphical, Numerical, Algebraic

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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