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Convexity domain of x cubed plus y cubed plus Axy

The hessian matrix of

f (x, y) = x^{3} + y^{3} + A x y

is positive semidefinite exactly when

x \geq 0, y \geq 0, 36 x y \geq A^{2} .

This region is convex: for

A \neq = 0

it is the epigraph of

A^{2} / (36 x)

in the positive quadrant, and for

A = 0

it is the closed first quadrant.

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