- #1
kostoglotov
- 234
- 6
Homework Statement
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
Homework Equations
The Attempt at a Solution
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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?