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I am having some difficulty with this question: $$\text{Prove that } u(x_1,x_2)=-x_1^2-2x_2^2+2x_1x_2-10x_1+40x_2 \text{represents a strictly convex preference and has a global satiation point}$$

I really don't know how to approach this problem. For the convex part, I think I need to prove that $u(x_1,x_2)$ is strictly quasi-concave, but I don't know how to do that.

Thank you very much.

Jackaba
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Convexity means that the quadratic term is negative definite. In this case, this means that the matrix

$$\begin{pmatrix} -1 & 1\\ 1 & -2\end{pmatrix}$$ is negative definite. Since the product of its eigenvalues (the determinant) is positive, and the sum (the trace) is negative, it is, so you are in luck.

Igor Rivin
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