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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 4.2, Problem 11E

a)

To determine

**To sketch:** The graph of a function thatsatisfies the conditions that the graph has local maximum at 2 and is differentiable at 2.

Expert Solution

Let the *x* be represented in the *x*-axis and the value of the function *y*-axis.

Since an absolute maximum occurs at the point 2, choose the greatest point on *x*-axis. That is, consider the point on (2, 2).

The graph of a differentiable function must have a tangent at each point in its domain. it should be a smooth curve and cannot contain any break or cusps.

Since the graph is differentiable at 2, the derivative of

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

From Figure 1, it is observed that the absolute maximum occurs at *x* = 2.

Also observe that the graph of the function is continuous and differentiable at *x* = 2.

b)

To determine

**To sketch:** The graph of a function that satisfies the conditions that the graph has local maximum at 2 and it is continuous but not differentiable at 2.

Expert Solution

Let the *x* be represented in the *x*-axis and the value of the function *y*-axis.

Since the local maximum occurs at the point 2, choose the point 2 at any open interval of a domain. That is, consider the point on (2, 2).

Since the graph is not differentiable at 2 the derivative of

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

From Figure 2 observe that the local maximum occurs at *x* = 2. And the curve is continuous.

Also, observe that the graph not differentiable at *x* = 2.

c)

To determine

**To sketch:** The graph of a function that satisfies the conditions that the graph has local maximum at 2 and it is not continuous at 2.

Expert Solution

Let the *x* be represented in the *x*-axis and the value of the function *y*-axis.

Since local maximum occurs at the point 2, choose the point 2 on the open interval of a domain. That is, consider the point on (2, 2).

A function is discontinuous at *x* = 2 if the graph of the function has break at that point.

Draw the graph of the function *f* in such a way that it satisfies the given conditions as shown below in Figure 3.

From Figure 3 observe that the local maximum occurs at *x* = 2 and the function is not continuous at *x* = 2.