BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter 4.2, Problem 12E

(a)

To determine

To sketch: The graph of a function on [1,2] that has an absolute maximum but no local maximum.

Expert Solution

Explanation of Solution

Let x be represented in the x-axis and the value of the function f(x) be represented in the y-axis.

Absolute maximum is any point of the domain at which the function attains its maximum value. Since the given interval is [1,2] , choose the maximum point at x = 2.

Local maximum is the point in the considered interval of the domain at which the function attains its maximum value.

Draw the graph of the function f in such a way that it satisfies the given conditions as shown below in Figure 1.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4.2, Problem 12E , additional homework tip  1

From Figure 1, observe that the absolute maximum occurs at x=2 . Since local extrema cannot occur at end points, the graph has no local maximum.

(b)

To determine

To sketch: The graph of a function on [1,2] and it satisfies the conditions that the graph has local maximum but no absolute maximum.

Expert Solution

Explanation of Solution

Let x be represented in the x-axis and the value of the function f(x) be represented in the y-axis.

Local maximum is the point in the considered interval of the domain at which the function attains its maximum value.

Consider any greatest point on the interval [1,2] but should not consider the end points as the local maximum does not at end points. Hence, consider the point (1, 1).

Absolute maximum is any point of the domain at which the function attains its maximum value.

Draw the graph of the function f in such a way that it satisfies the given conditions as shown below in Figure 2.

Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter 4.2, Problem 12E , additional homework tip  2

From Figure 2, it is observed that the graph has local maximum at x=1 . Since the curve has hole at x=1 , the point is not included in the domain. Therefore, it has no absolute maximum at x=1 .

Note: If it has no hole at x=1 then absolute maximum occurs at x=1 .

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