Question

Hi, this question is about Langrange multipliers.

Given f(x,y) = y^2 - x^2, subject to the constraint g(x,y) = 0.25x^2 + y^2 = 1, find the max and mins.

So I found the partial derivatives for both f(x,y) and g(x,y):

fx = -2x

fy = 2y

gx = 0.5x

gy = 2y

And set them up for the Lagrange multipliers method. In essence, grad f(x,y) = lambda times grad g(x,y).

My algebra lead me to the roots of x=0 from the x partial derivatives and y=0 from the y partial derivatives. I plugged these into g(x,y) and got the points (0, ±1) (±2, 0).

So the book agrees with my critical points being (0, ±1) (±2, 0). The book also agrees that (0, ±1) should both be a max point equal to z=1.

However, the book says (2, 0) and (-2, 0) should both be a min point at z=4. Now if I'm not mistaken, to be a min point you must have fxx(a, b) > 0, but that can't be possible if fxx = d(fx)/dx = (-2x)' = -2. Also, it says f(±2, 0) = -4. Wouldn't that also require the x root being ±16?

Any insight appreciated.