Chapter 14, Problem 58RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use a graphing calculator or computer (or Newton’s method or a computer algebra system) to find the critical points of *f*(*x*, *y*) = 12 + 10*y* – 2*x*^{2} – 8*xy* − *y*^{4} correct to three decimal places. Then classify the critical points and find the highest point on the graph.

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),

Substitute

Find the value of

Use online graphing calculator and draw the graph of

From Figure 1 it can be observed that the values of

Substitute

Substitute

Substitute

Therefore, the critical points of the function

Find the second derivatives as follows.

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

Hence,

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