Is this function a convex function?

[mohit]

Convex Functions

Could anyone comment on the convexity of

f(x,y) = (x^2) * exp ...... i.e. x square into e to the power y.

I did try to find Hessian of the same and the value I get is :

Hessian(x,y) = -2 * x^2 * exp(2y).... which looks <= 0 for all x and y.

I assume this should imply f(x,y) is concave. However when I plot this function using a 3D graph plotter it seems convex .

try
https://www.livephysics.com/ptools/online-3d-function-grapher.php

for plotting function.

I am sure I am making a simple mistake or something.

Any help would be useful.

 

[htg]

Re: Convex Functions

f(x,y)= x^2 * exp is not convex (you can verify that if you connect points (1,10,f(1,10)) and (10,6,f(10,6)) then the center of this segment lies below the graph of f).
A function is convex if the Hessian matrix is positive definite - it is not enough to have positive determinant of the Hessian matrix.
In your case the Hessian matrix is not positive definite because the top left corner is positive, but the determinant of the whole matrix is <= 0.
See Wikipedia on Positive definite matrix, and on Hessian.