Determine whether the convex, concave, or neither.

Question

$$ \ f(x, y)= \ 5^2 + 2^2 − 3 + \\ on\ S=\mathbb{R}^2 $$

Hello, I can determine only $f(x)$ functions whether convex or concave but I could not solve problem like this $f(x,y)$, partial derivative. Could someone help me about this problem? By the way, it is my first question in this site! I'm glad to be here.

here is another question like above:

$$ \ f(x, y)= \ ^{-} + ^2 − 2 \\ on\ S=\mathbb{R}^2 $$

Thanks in advance!:)

Answer 1

You need to get the Hessian matrix of $f$ first (https://en.wikipedia.org/wiki/Hessian_matrix), then check if the Hessian is positive definite or negative definite. If it's positive definite, then $f$ is strict convex. If it's positive semi-definite, then $f$ is convex.

For the first $f$ you provide, $\frac{\partial f}{\partial x}=10x-3y$, $\frac{\partial f}{\partial y}=4y-3x+1$. Follow the definition of Hessian in https://en.wikipedia.org/wiki/Hessian_matrix. $H= \begin{bmatrix} 10 & -3 \\ -3 & 4 \end{bmatrix}$. Since H is positive definite, we conclude that f is strict convex over $(x,y)$.

we can go through the similar steps for the second equation.