Chapter 14.4, Problem 3CP

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

Chapter
Section

### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Suppose that z = 4 − x 2 − y 2 + 2 x − 4 y .Test the point(s) for relative maxima and minima.

To determine

To calculate: The relative maxima and relative minima of the function z=4x2y2+2x4y, if they exist.

Explanation

Given Information:

The provided function is, z=4âˆ’x2âˆ’y2+2xâˆ’4y.

Formula used:

To calculate relative maxima and minima of the z=f(x,y),

1. Find the partial derivatives âˆ‚zâˆ‚x and âˆ‚zâˆ‚y.

2. Find the critical points, that is, the point(s) that satisfy âˆ‚zâˆ‚x=0 and âˆ‚zâˆ‚y=0.

3. Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)âˆ’(zxy)2=âˆ‚2zâˆ‚x2â‹…âˆ‚2zâˆ‚y2âˆ’(âˆ‚2zâˆ‚xâˆ‚y)2.

(a) If D>0, then relative minimum occurs if zxx>0 and relative maximum occurs if zxx<0.

(b) If D<0, then neither a relative maximum nor a relative minimum occurs.

For a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant and the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to x is denoted by fx and with respect to y is denoted by fy.

For a function z(x,y), the second partial derivative,

(1) When both derivatives are taken with respect to x is zxx=âˆ‚2zâˆ‚x2=âˆ‚âˆ‚x(âˆ‚zâˆ‚x).

(2) When both derivatives are taken with respect to y is zyy=âˆ‚2zâˆ‚y2=âˆ‚âˆ‚y(âˆ‚zâˆ‚y).

(3) When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=âˆ‚2zâˆ‚yâˆ‚x=âˆ‚âˆ‚y(âˆ‚zâˆ‚x).

(4) When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=âˆ‚2zâˆ‚xâˆ‚y=âˆ‚âˆ‚x(âˆ‚zâˆ‚y).

Power of x rule for a real number n is such that, if f(x)=xn then fâ€²(x)=nxnâˆ’1.

Constant function rule for a constant c is such that, if f(x)=c then fâ€²(x)=0.

Coefficient rule for a constant c is such that, if f(x)=câ‹…u(x), where u(x) is a differentiable function of x, then fâ€²(x)=câ‹…uâ€²(x).

Calculation:

Consider the function, z=4âˆ’x2âˆ’y2+2xâˆ’4y.

Recall that, for a function f(x,y), the partial derivative of f with respect to x is calculated by taking the derivative of f(x,y) with respect to x and keeping the other variable y constant and the partial derivative of f with respect to y is calculated by taking the derivative of f(x,y) with respect to y and keeping the other variable x constant.

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

Thus,

âˆ‚zâˆ‚x=âˆ’2x+2

And,

âˆ‚zâˆ‚y=âˆ’2yâˆ’4

Set âˆ‚zâˆ‚x=0 and âˆ‚zâˆ‚y=0 to find the critical point.

âˆ‚zâˆ‚x=0âˆ’2x+2=0x=1

And,

âˆ‚zâˆ‚y=0âˆ’2yâˆ’4=0y=âˆ’2

Substitute 1 for x and âˆ’2 for y in z=4âˆ’x2âˆ’y2+2xâˆ’4y

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started