Section 14.8: Lagrange Multipliers

The method of Lagrange multipliers assumes that the extreme values exist, but that is not always the case. Show that the problem of finding the minimum value of \(f(x, y) = x\) subject to the constraint \(y^2 + x^4 - x^3 = 0\) can be solved using Lagrange multipliers, but \(f\) does not have a maximum value with that constraint.