Chapter 14.7, Problem 28E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

Use a graphing device as in Example 4 (or Newton’s method or solve numerically using a calculator or computer) to find the critical points of *f* correct to three decimal places. Then classify the critical points and find the highest or lowest points on the graph, if any.

**28.** *f*(*x*, *y*) = *y*^{6} − 2*y*^{4}+ *x*^{2} − *y*^{2} + *y*

To determine

**To find:** The critical points of the function

Explanation

**Result used:**

**Second Derivative Test:**

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

Let

**Given:**

The function is,

**Calculation:**

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 (1),

From equation (2),

Find the value of

Use online graphing calculator and draw the graph of

From Figure 1 it can be observed that the values of

Therefore, the critical points of the function

Obtain the second derivatives as follows.

Take the partial derivative of the equation (1) with respect to *x* and obtain

Hence,

Take the partial derivative of the equation (2) with respect to *y* and obtain

Hence,

Take the partial derivative of the equation (1) with respect to *y* and obtain

Hence,

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