   Chapter 14.5, Problem 2CP 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

Find the minimum value of f ( x , y ) = x 2 + y 2 − 4 x y , subject to the constraint x + y = 10 , byfinding ∂ F ∂ x , ∂ F ∂ y ,  and  ∂ F ∂ λ ,

To determine

To calculate: The partial derivatives Fx, Fy and Fλ of the objective function F(x,y,λ) of the function f(x,y)=x2+y24xy subject to the constraint x+y=10.

Explanation

Given Information:

The provided function is f(x,y)=x2+y24xy subject to the constraint x+y=10.

Formula used:

According to the Lagrange multipliers method to obtain maxima or minima for a function the objective function F(x,y,λ) of a function f(x,y) subject to the constraint g(x,y)=0 is given by F(x,y,λ)=f(x,y)+λg(x,y).

For a function f(x,y), 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 y is denoted by fy.

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

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, f(x,y)=x2+y24xy

The provided constraint is x+y=10.

According to the Lagrange multipliers method,

The objective function is F(x,y,λ)=f(x,y)+λg(x,y).

Thus, f(x,y)=x2+y24xy and g(x,y)=x+y10.

Substitute x2+y24xy for f(x,y) and x+y10 for g(x,y) in F(x,y,λ)=f(x,y)+λg(x,y)

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