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Mathematical Applications for the ...

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

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BuyFindarrow_forward

Mathematical Applications for the ...

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

Let z = 6 x 2 + x 2 y + y 2 4 y +   9 . Find the pairs ( x , y ) that are critical points for z and then classify each as a relative maximum, a relative minimum, or a saddle point.

To determine

To calculate: The pairs (x,y) that are critical points for z and then classify each as a relative maxima, relative minima, and saddle points of z=6x2+x2y+y24y+9.

Explanation

Given Information:

The provided function is, z=6x2+x2y+y24y+9.

Formula used:

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

1. Find the partial derivatives zx and zy.

2. Find the critical points, that is, the point(s) that satisfy zx=0 and zy=0.

3. Then find all the second partial derivatives and evaluate the value of D at each critical point, where D=(zxx)(zyy)(zxy)2=2zx22zy2(2zxy)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=2zx2=x(zx).

2. When both derivatives are taken with respect to y is zyy=2zy2=y(zy).

3. When first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx).

4. When first derivative is taken with respect to y and second derivative is taken with respect to x is zyx=2zxy=x(zy).

Power of x rule for a real number n is such that, if f(x)=xn then f(x)=nxn1.

Chain rule for function f(x)=u(v(x)) is f(x)=u(v(x))v(x).

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

Coefficient rule for a constant c is such that, if f(x)=cu(x), where u(x) is a differentiable function of x, then f(x)=cu(x).

Calculation:

Consider the function, z=6x2+x2y+y24y+9.

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, the chain rule, and the coefficient rule,

Thus,

zx=012x+2xy=02x(6+y)=0

Thus, x=0 or y=6.

And,

zy=0x2+2y4=0

Consider the equation, x=0

Substitute 0 for x in x2+2y4=0.

0+2y4=0y=2

Consider the equation, y=6

Substitute 6 for y in x2+2y4=0.

x2+2(6)4=0x2124=0x2=16x=±4

Thus, the critical points are (0,2), (4,6) and (4,6).

Recall that, for a function z(x,y), the second partial derivative, when both derivatives are taken with respect to x is zxx=2zx2=x(zx), when both derivatives are taken with respect to y is zyy=2zy2=y(zy), when first derivative is taken with respect to x and second derivative is taken with respect to y is zxy=2zyx=y(zx)

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Chapter 14 Solutions

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Sect-14.1 P-10ESect-14.1 P-11ESect-14.1 P-12ESect-14.1 P-13ESect-14.1 P-14ESect-14.1 P-15ESect-14.1 P-16ESect-14.1 P-17ESect-14.1 P-18ESect-14.1 P-19ESect-14.1 P-20ESect-14.1 P-21ESect-14.1 P-22ESect-14.1 P-23ESect-14.1 P-24ESect-14.1 P-25ESect-14.1 P-27ESect-14.1 P-28ESect-14.1 P-29ESect-14.1 P-30ESect-14.1 P-31ESect-14.1 P-32ESect-14.1 P-33ESect-14.1 P-34ESect-14.1 P-35ESect-14.1 P-36ESect-14.1 P-37ESect-14.1 P-38ESect-14.2 P-1CPSect-14.2 P-2CPSect-14.2 P-3CPSect-14.2 P-4CPSect-14.2 P-5CPSect-14.2 P-1ESect-14.2 P-2ESect-14.2 P-3ESect-14.2 P-4ESect-14.2 P-5ESect-14.2 P-6ESect-14.2 P-7ESect-14.2 P-8ESect-14.2 P-9ESect-14.2 P-10ESect-14.2 P-11ESect-14.2 P-12ESect-14.2 P-13ESect-14.2 P-14ESect-14.2 P-15ESect-14.2 P-16ESect-14.2 P-17ESect-14.2 P-18ESect-14.2 P-19ESect-14.2 P-20ESect-14.2 P-21ESect-14.2 P-22ESect-14.2 P-23ESect-14.2 P-24ESect-14.2 P-25ESect-14.2 P-26ESect-14.2 P-27ESect-14.2 P-28ESect-14.2 P-29ESect-14.2 P-30ESect-14.2 P-31ESect-14.2 P-32ESect-14.2 P-33ESect-14.2 P-34ESect-14.2 P-35ESect-14.2 P-36ESect-14.2 P-37ESect-14.2 P-38ESect-14.2 P-39ESect-14.2 P-40ESect-14.2 P-41ESect-14.2 P-42ESect-14.2 P-43ESect-14.2 P-44ESect-14.2 P-45ESect-14.2 P-46ESect-14.2 P-47ESect-14.2 P-48ESect-14.2 P-49ESect-14.2 P-50ESect-14.2 P-51ESect-14.2 P-52ESect-14.2 P-53ESect-14.2 P-54ESect-14.2 P-55ESect-14.2 P-56ESect-14.3 P-1CPSect-14.3 P-2CPSect-14.3 P-3CPSect-14.3 P-1ESect-14.3 P-2ESect-14.3 P-3ESect-14.3 P-4ESect-14.3 P-5ESect-14.3 P-6ESect-14.3 P-7ESect-14.3 P-8ESect-14.3 P-9ESect-14.3 P-10ESect-14.3 P-11ESect-14.3 P-12ESect-14.3 P-13ESect-14.3 P-14ESect-14.3 P-15ESect-14.3 P-16ESect-14.3 P-17ESect-14.3 P-18ESect-14.3 P-19ESect-14.3 P-20ESect-14.3 P-21ESect-14.3 P-22ESect-14.3 P-23ESect-14.3 P-24ESect-14.3 P-25ESect-14.3 P-26ESect-14.3 P-27ESect-14.3 P-28ESect-14.3 P-29ESect-14.3 P-30ESect-14.4 P-1CPSect-14.4 P-2CPSect-14.4 P-3CPSect-14.4 P-4CPSect-14.4 P-1ESect-14.4 P-2ESect-14.4 P-3ESect-14.4 P-4ESect-14.4 P-5ESect-14.4 P-6ESect-14.4 P-7ESect-14.4 P-8ESect-14.4 P-9ESect-14.4 P-10ESect-14.4 P-11ESect-14.4 P-12ESect-14.4 P-13ESect-14.4 P-14ESect-14.4 P-15ESect-14.4 P-16ESect-14.4 P-17ESect-14.4 P-18ESect-14.4 P-19ESect-14.4 P-20ESect-14.4 P-21ESect-14.4 P-22ESect-14.4 P-23ESect-14.4 P-24ESect-14.4 P-25ESect-14.4 P-26ESect-14.4 P-27ESect-14.4 P-28ESect-14.4 P-29ESect-14.4 P-30ESect-14.4 P-31ESect-14.4 P-32ESect-14.4 P-34ESect-14.4 P-35ESect-14.4 P-36ESect-14.5 P-1CPSect-14.5 P-2CPSect-14.5 P-3CPSect-14.5 P-4CPSect-14.5 P-1ESect-14.5 P-2ESect-14.5 P-3ESect-14.5 P-4ESect-14.5 P-5ESect-14.5 P-6ESect-14.5 P-7ESect-14.5 P-8ESect-14.5 P-9ESect-14.5 P-10ESect-14.5 P-11ESect-14.5 P-12ESect-14.5 P-13ESect-14.5 P-14ESect-14.5 P-15ESect-14.5 P-16ESect-14.5 P-17ESect-14.5 P-18ESect-14.5 P-19ESect-14.5 P-20ESect-14.5 P-21ESect-14.5 P-22ESect-14.5 P-23ESect-14.5 P-24ESect-14.5 P-25ESect-14.5 P-26ECh-14 P-1RECh-14 P-2RECh-14 P-3RECh-14 P-4RECh-14 P-5RECh-14 P-6RECh-14 P-7RECh-14 P-8RECh-14 P-9RECh-14 P-10RECh-14 P-11RECh-14 P-12RECh-14 P-13RECh-14 P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-1TCh-14 P-2TCh-14 P-3TCh-14 P-4TCh-14 P-5TCh-14 P-6TCh-14 P-7TCh-14 P-8TCh-14 P-9TCh-14 P-10T

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