Hello,

My question is :

Show that the function

$\displaystyle f(x,y) = xy$ has a saddle point at $\displaystyle (x,y) = (0,0)$ and find the second derivative along a path through that point has the greatest positive and negative values.

My thoughts:

If I substitute y = x into the function as thus $\displaystyle f(y,y) = x.x = x^2$ and then find the second derivative as thus $\displaystyle d^2f/dx^2 = 2$ this is repeated for the path y = -x as thus $\displaystyle f(y,y) = -x.x = -x^2$with a second derivative of $\displaystyle d^2f/dx^2 = -2$

This is repeated for paths x=y and x=-y giving a derivative of $\displaystyle d^2f/dy^2 = 2$ and $\displaystyle d^2f/dy^2 = -2$ repectively. So the greatest positive and negative values are 2 and -2. At y=0 and x = 0 the function is constant.

Any Help would be greatly appreciated.

Thank you,

Riptorn70.