   Chapter 14.8, Problem 25E

Chapter
Section
Textbook Problem

Consider the problem of minimizing the function f(x, y) = x on the curve y2 + x4 - x3 = 0 (a piriform).(a) Try using Lagrange multipliers to solve the problem.(b) Show that the minimum value is f(0, 0) = 0 but the Lagrange condition ∇f(0, 0) = λ∇g (0, 0) is not satisfied for any value of λ.(c) Explain why Lagrange multipliers fail to find the minimum value in this case.

(a)

To determine

To find: The extreme value of the function f(x,y)=x subject to the constraint y2+x4x3=0 by using Lagrange multipliers.

Explanation

Given:

The function is f(x,y)=x subject to the constraint y2+x4x3=0 .

Result used:

“The Lagrange multipliers defined as f(x,y,z)=λg(x,y,z) . This equation can be expressed as fx=λgx , fy=λgy , fz=λgz and g(x,y,z)=k ”.

Calculation:

The given function is, f(x,y)=x and g(x,y)=y2+x4x3 .

The Lagrange multipliers f(x,y)=λg(x,y) is computed as follows,

f(x,y)=λg(x,y)fx,fy=λgx,gyfx(x),fy(x)=λgx(y2+x4x3),gy(y2+x4x3)1,0=λ4x

(b)

To determine

To show: The minimum value of f(0,0)=0 and find the Lagrange condition f(0,0)=λg(0,0) does not exist.

(c)

To determine

To explain: The failure of the Lagrange multipliers while finding the minimum values.

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