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how can I determine if the conditioned function is convex? If so, how can I prove it?

The function is $$ f(x,y)=\frac{x^2}{y^2}+y^2,\\y\geq\sqrt{x}\quad\text {and} \quad x>0, $$ That's the question, I can't tell if it's convex.[![function figure over [0.1, 2]$\times$[0.1, 2] region,

Jack
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  • The region is not convex so the question of convexity of the function does not arise. – Kavi Rama Murthy Apr 09 '20 at 09:07
  • But given any x, f(x,y) is increasing, and given any y, it is increasing, and both are convex,I don't understand why the whole is not. – Jack Apr 09 '20 at 09:16

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The line joining two points of the domain does not lie in the domain so the domain is not convex. Consequently your function is not convex. enter image description here

  • Yes. you are right. I need to modify the condition. I forget the equality. – Jack Apr 09 '20 at 09:25
  • Sorry, for my mistake. – Jack Apr 09 '20 at 09:27
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    @Jack: Even with equality it is not going to work. Just pick two points on the lower border. Draw a line, it will not lie in the red region. –  Apr 09 '20 at 09:31