Chapter 8.3, Problem 40E

To determine

Whether the statement “If $f\left(x,y\right)$ satisfies $f_{xx}\left(a,b\right)\ne 0$ , $f_{xy}\left(a,b\right)=0$ and $f_{yy}\left(a,b\right)\ne 0$ and $f_{xx}\left(a,b\right)+f_{yy}\left(a,b\right)=0$ at the critical point $\left(a,b\right)$ of f , then f cannot have a relative extremum.” is true or false. Explain the reason if it is true or given an example if it is false.