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.