Chapter 14, Problem 12RQ

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

**12.** If *f*(*x*, *y*) has two local maxima, then *f* must have a local minimum.

To determine

Whether the statement “If

Explanation

**Second Derivative Test:**

“Suppose the second partial derivatives of *f* are continuous on a disk with center
*f*).

Let

**Reason:**

The given statement is false since the below example disproves the given statement.

If the function

Take the partial derivative in the given function with respect to *x* and obtain

Thus,

Take the partial derivative in the given function with respect to *y* and obtain

Thus,

Solve the equations (1) and (2) and find the values of *x* and *y*.

From the equation (2),

From the result there are two possibilities such as either

Suppose

Thus, there are no critical points for

Therefore,

Thus, the value of

Therefore, the critical points of the function

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