Consider the problem of minimizing the function f(x, y) = x on the curve y^2 + x^4 − x^3 = 0 (a type of curve known as a "piriform").

(a) Try using Lagrange multipliers to solve the problem.
(b) Show that the minimum value is f(0, 0) but the Lagrange condition ∇f(0, 0) = λ∇g(0, 0) is not satisfied for any value λ.
(c) Explain why Lagrange multipliers fail to find the minimum value in this case.

(Hint for parts (b) and (c): try to plot the curve using Wolfram Alpha, and locate where the point (0, 0) is on this curve.)