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Check if the function below is concave.

$$f(x,y) = (x-2)(2000+4\sqrt y-20x) - 10000 - y$$

I tried finding the eigenvalue for the Hessian where

$$det(H-\lambda) = det \begin{pmatrix} -40-\lambda & \frac{2}{\sqrt y} \\ \frac{2}{\sqrt y} & -\frac{x-2}{y^{3/2}} - \lambda \end{pmatrix} = 0 $$ but I got stuck because the eigenvalue is dependent on x and y.

Is there another way to do this, aside from using the definition of concavity?

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    This is neither convex nor concave. And since it only takes one counterexample to prove that, just select a particular $x,y$ and compute the eigenvalues of $H$ for that. – 2017-02-09

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