| # Convexity of quadratic functions in two variables We return to the example described here. We consider the two quadratic functions , with values The Hessian of is independent of , and given by the constant matrix: We check that the eigenvalues of are positive, since the determinant, as well as the trace, of the above matrix are. Therefore, is convex. Likewise, the Hessian of is This time, the Hessian has a negative eigenvalue, so is not convex.
| Level sets and graph of the quadratic function . The epigraph is anything that extends above the graph in the -axis direction. This function is ‘‘bowl-shaped’’, or convex.
| Level sets and graph of the quadratic function .