Chapter 13 - Section 13.7 - Extreme Values and Saddle Points - Exercises - Page 737: 23

Saddle point at $(n \pi,0)$

Since, we have $f_x(x,y)=y \cos x=0, f_y(x,y)=\sin x=0$ After solving the above two equations, we get The critical point is: $(n \pi,0)$ As per the second derivative test, for $(n \pi,0)$ we have $D=f_{xx}f_{yy}-f^2_{xy}=-1$ and $D=-1 \lt 0$ Thus, we have: Saddle point at $(n \pi,0)$