Task: Show that the function x^2-2xy+5y^2-10x+2y is convex. Find its global minimum value.

Question

This is a task for economics calculus, and I cannot do it.

Answer 1

The minimum of this function is at the point (6,1)  Changing variables to x' = x -6  and y' -1  results in

 

f(x', y' ) =  x'²  - 2 x' y' + 5 y'² +24

 

It is possible to show that x'² - 2 x' y' + 5 y'²     is always greater than zero.  

This implies that the function is convex.