Chapter 14, Problem 3T

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11th Edition

Ronald J. Harshbarger + 1 other

ISBN: 9781305108042

Textbook Problem

Let
*z* and then classify each as a relative maximum, a relative minimum, or a saddle point.

To determine

**To calculate:** The pairs *z* and then classify each as a relative maxima, relative minima, and saddle points of

Explanation

**Given Information:**

The provided function is,

**Formula used:**

To calculate relative maxima and minima of the

1. Find the partial derivatives

2. Find the critical points, that is, the point(s) that satisfy

3. Then find all the second partial derivatives and evaluate the value of *D* at each critical point, where

(a) If

(b) If

For a function *x* is calculated by taking the derivative of *x* and keeping the other variable *y* constant and the partial derivative of *y* is calculated by taking the derivative of *y* and keeping the other variable *x* constant. The partial derivative of *x* is denoted by *y* is denoted by

For a function

1. When both derivatives are taken with respect to *x* is

2. When both derivatives are taken with respect to *y* is

3. When first derivative is taken with respect to *x* and second derivative is taken with respect to *y* is

4. When first derivative is taken with respect to *y* and second derivative is taken with respect to *x* is

Power of

Chain rule for function

Constant function rule for a constant

Coefficient rule for a constant

**Calculation:**

Consider the function,

Recall that, for a function *x* is calculated by taking the derivative of *x* and keeping the other variable *y* constant and the partial derivative of *y* is calculated by taking the derivative of *y* and keeping the other variable *x* constant.

Use the power of *x* rule for derivatives, the constant function rule, the chain rule, and the coefficient rule,

Thus,

Thus,

And,

Consider the equation,

Substitute

Consider the equation,

Substitute

Thus, the critical points are

Recall that, for a function *x* is *y* is *x* and second derivative is taken with respect to *y* is

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