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Question: Consider the intersection of two surfaces: an elliptic paraboloid z=x^2 + 2*x + 4*y^2 and a right circular cylinder x^2 + y^2 = 1. Use Lagrange multipliers to find the highest and lowest points on the curve of the intersection.

What I have so far:

I managed to find my critical points using lagrange multipliers. But now I don't know how to describe whether my points are at maximum or minimum...

The points I found were: (2/3, -[tex]\sqrt{5}[/tex]/3) and (-[tex]\sqrt{2}[/tex]/6, 1)