Chapter 14, Problem 9T

### 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

# Find x and y that maximize the utility function U   = x 3 y subject to the budget constraint 30 x   +   20 y =   8000 .

To determine

To calculate: The values of x and y which maximizes the utility function U=x3y and subject to the budget constraint 30x+20y=8000.

Explanation

Given Information:

The provided utility function is U=x3y and the constraint 30x+20y=8000.

Formula used:

According to the Lagrange multipliers method to obtain maxima or minima for a function z=f(x,y) subject to the constraint g(x,y)=0,

(1) Find the critical values of f(x,y) using the new variable Î» to form the objective function F(x,y,Î»)=f(x,y)+Î»g(x,y).

(2) The critical points of f(x,y) are the critical values of F(x,y,Î») which satisfies g(x,y)=0.

(3) The critical points of F(x,y,Î») are the points that satisfy âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, and âˆ‚Fâˆ‚Î»=0, that is, the points which make all the partial derivatives of zero.

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 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, U=x3y.

The provided constraint is 30x+20y=8000.

According to the Lagrange multipliers method,

The objective function is F(x,y,Î»)=f(x,y)+Î»g(x,y).

Thus, f(x,y)=x3y and g(x,y)=30x+20yâˆ’8000.

Substitute x3y for f(x,y) and 30x+20yâˆ’8000 for g(x,y) in F(x,y,Î»)=f(x,y)+Î»g(x,y).

F(x,y,Î»)=x3y+Î»(30x+20yâˆ’8000)

Since, the critical points of F(x,y,Î») are the points that satisfy âˆ‚Fâˆ‚x=0, âˆ‚Fâˆ‚y=0, and âˆ‚Fâˆ‚Î»=0.

Recall that, 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.

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

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