Consider the problem of minimizing (x, y) = x subject to g(x, y) = (x 1)

Consider the problem of minimizing ƒ(x, y) = x subject to g(x, y) = (x − 1)³ − y² = 0.

(a) Show, without using calculus, that the minimum occurs at P = (1, 0).
(b) Show that the Lagrange condition ∇ƒP = λ∇gP is not satisfied for any value of λ.
(c) Does this contradict Theorem 1?

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THEOREM 1 Lagrange Multipliers Assume that f(x, y) and g(x, y) are differen- tiable functions. If f(x, y) has a local minimum or a local maximum on the constraint curve g(x, y) = 0 at P = (a, b), and if Vgp #0, then there is a scalar such that Vfp = Vgp