1. The problem statement, all variables and given/known data My question is quite specific, but I will include the entire question as laid out in the text Consider the problem of minimizing the function f(x,y) = x on the curve y^2 + x^4 -x^3 = 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 [itex]\nabla f(0,0) = \lambda \nabla g(0,0)[/itex] is not satisfied for any value of [itex]\lambda[/itex] (c) Explain why Lagrange Multipliers fail to find the minimum values in this case 2. Relevant equations 3. The attempt at a solution I've answered (c) correctly, but I'm not happy with my own answer, because I don't really understand why it's correct. I arrived at the answer by plotting f(x,y) = x in Matlab with contour curves of the constraint and then zoomed in on the contour curve where it equals 0. I got this: It's hard to see, but where that red ring is, is (0,0,0), which is the constrained min of f(x,y). So I can see graphically that my constraint is discontinuous at (0,0). The solutions manual to the text gives the answer as "[itex]\nabla g(0,0) = 0[/itex] and one of the conditions of the Lagrange method is that [itex]\nabla g(x,y) \neq 0[/itex]". Ok, so a condition of the method is that the grad vector of the constraint not be a zero vector. But why? I tried solving the general form of the constraint as a limit as (x,y) approach (0,0) but couldn't get an answer. Yet I can clearly see on the graph that the constraint as a level curve at g(x,y) = 0 is discontinuous at (0,0). If I hadn't had Matlab available, I wouldn't have been able to answer this question. How could I have approached it analytically?