Chapter 14, Problem 20RE

### 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

# Test z = x 3 + y 3 −   12 x   − 27 y for maxima and minima.

To determine

To calculate: The relative maxima, relative minima and saddle points of z=x3+y312x27y if they exist.

Explanation

Given Information:

The provided function is, z=x3+y312x27y.

Formula used:

Do the following steps to calculate relative maxima and minima of the function 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=x3+y312x27y.

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=03x212=0x2=4x=±2

And,

zy=03y227=0y2=9y=±3

Thus, the critical points are (2,3), (2,3), (2,3) and (2,3).

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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