Use Lagrange multipliers to find the maximum and minimum values of f(x; y) = x^2+4y^3 subject to the constraint x^2 + 2y^2 = 8. Also, find the points at which these extreme values occur.
Using Lagrange multipliers, we get, 2x = λ2x
12y^2 = λ4y
From the first equation, we get λ=1, putting in the second equation we get y=1/3, 0. Using these two values of y and constraint equation, we get x = √70/3,2√2) respectively. Thus the points at which maxima and minima occur are (√70/3,1/3) and (2√2,0) but the actual minima is at (0,-2). Am I doing something wrong?