   Chapter 14.5, 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

Find the maximum value of w   = x 2 y z subject to the constraint 4 x + y   +   z =   4 , x   ≥ 0 , y   ≥ 0 ,  and  z ≥ 0 .

To determine

To calculate: The maximum value of w=x2yz which is subjected to the constraint 4x+y+z=4, x0, y0 and z0.

Explanation

Given Information:

The provided function is w=x2yz and it is subjected to the constraint 4x+y+z=4, x0, y0 and z0.

Formula used:

Lagrange Multipliers Method:

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

Step 1: Find the critical values of f(x,y,z) using the new variable λ to form the objective function F(x,y,z,λ)=f(x,y,z)+λg(x,y,z).

Step 2: The critical points of f(x,y,z) are the critical values of F(x,y,z,λ) which satisfies g(x,y,z)=0.

Step 3: The critical points of F(x,y,z,λ) are the points that satisfy:

Fx=0, Fy=0, Fz=0 and Fλ=0, that is, the points which make all the partial derivatives equal to zero.

For a function f(x,y,z), the partial derivative of f(x,y,z) with respect to y is calculated by taking the derivative of f(x,y,z) with respect to y and keeping the other variables x and z constant. The partial derivative of f(x,y,z) 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, w=x2yz.

The provided constraint is 4x+y+z=4, x0, y0 and z0.

According to the Lagrange multipliers method,

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

Here, f(x,y,z)=x2yz and g(x,y,z)=4x+y+z4.

Put the values of f(x,y,z)=x2yz and g(x,y,z)=4x+y+z4 in the objective function, F(x,y,z,λ)=f(x,y,z)+λg(x,y,z).

F(x,y,z,λ)=x2yz+λ(4x+y+z4)

Since, the critical points of F(x,y,λ) are the points that satisfy:

Fx=0, Fy=0, Fz=0 and Fλ=0.

Recall that, for a function f(x,y,z), the partial derivative of f(x,y,z) with respect to y is calculated by taking the derivative of f(x,y,z) with respect to y and keeping the other variables x and z constant

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