James Stewart

ISBN: 9781305266643

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.

To determine

To find: The extreme value of the function f(x,y)=x subject to the constraint y2+x4−x3=0 by using Lagrange multipliers.

The function is f(x,y)=x subject to the constraint y2+x4−x3=0 .

Result used:

“The Lagrange multipliers defined as ∇f(x,y,z)=λ∇g(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+x4−x3 .

The Lagrange multipliers ∇f(x,y)=λ∇g(x,y) is computed as follows,

∇f(x,y)=λ∇g(x,y)〈fx,fy〉=λ〈gx,gy〉〈fx(x),fy(x)〉=λ〈gx(y2+x4−x3),gy(y2+x4−x3)〉〈1,0〉=λ〈4x

To determine

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

To determine

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

Check out a sample textbook solution.

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