   Chapter 14.4, Problem 14E Mathematical Applications for the ...

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

Solutions

Chapter
Section Mathematical Applications for the ...

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

In Problems 1-16, find each function’s relative maxima, relative minima, and saddle points, if they exist. z = x 2 + 5 x y + 10 y 2 + 8 x − 40 y

To determine

To calculate: The relative maxima, relative minima, and saddle points of z=x2+5xy+10y2+8x40y, if they exist.

Explanation

Given Information:

The provided function is z=x2+5xy+10y2+8x40y.

Formula used:

Differentiate z=f(x,y) with respect to x by holding y constant as:

x[f(x,y)]=fx(x,y)

Differentiate f(x,y) with respect to y by holding x constant as:

y[f(x,y)]=fy(x,y)

The every differentiable function f(x,y),

2yxf(x,y)=2xyf(x,y)

The following procedure are used to calculate relative maximum and saddle point of given function f(x,y).

Step-1: Calculate first derivative of function f(x,y) with respect to x and y.

Step-2 Calculate second derivative of function f(x,y) with respect to x and y.

Step-3: Equate first derivative of function f(x,y) with respect to x to zero.

Step-4: Equate first derivative of function f(x,y) with respect to y to zero.

Step-5: Calculate critical point of function f(x,y) that is (a,b).

Step-6: Now test function f(x,y) at critical point (a,b) for extrema and saddle point that is describe in following table.

 Critical point ∂2∂2xf(x,y) ∂2∂x2f(x,y)∂2∂y2f(x,y)−[∂2∂y∂xf(x,y)]2 Conclusion (a,b) ∂2∂2xf(a,b)>0 ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2>0 The function f(x,y) has relative minimum at point (a,b) (a,b) ∂2∂2xf(a,b)<0 ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2>0 The function f(x,y) has relative maximum at point (a,b) (a,b) ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2<0 The function f(x,y) has saddle point (a,b,f(a,b)) (a,b) ∂2∂x2f(a,b)∂2∂y2f(a,b)−[∂2∂y∂xf(a,b)]2=0 The test gives no information.

The simple power rule of derivative:

ddx(xn)=nxn1

The constant rule of derivative:

ddx(c)=0

The derivative of constant coefficient:

ddx(cf(x))=cf(x)

Calculation:

Consider the function, z=x2+5xy+10y2+8x40y.

Differentiate z=f(x,y) with respect to x by holding y constant,

zx=x(x2+5xy+10y2+8x40y)=2x+5y+8

Equate the first derivative of function z=x2+5xy+10y2+8x40y with respect to x to zero.

2x+5y+8=02x+5y=8

The first partial derivative of function z=x2+5xy+10y2+8x40y with respect to y by holding x constant

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