## Calculus 8th Edition

Minimum: $f(0,1.211) \approx -1.403$, $f(0,-1.273) \approx -3.890$ Saddle points at $(0,0.347)$ Lowest points: $(0,-1.273, -3.890)$
Second derivative test: Some noteworthy points to calculate the local minimum, local maximum and saddle point of $f$. 1. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\gt 0$ , then $f(p,q)$ is a local minimum. 2.If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\lt 0$ , then $f(p,q)$ is a local maximum. 3. If $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \lt 0$ , then $f(p,q)$ is a not a local minimum and local maximum or, a saddle point. Critical points are: $(0,1.211),(0,-1.273),(0,0.347),(0,-1.273)$ For $(x,y)=(0,-1.273)$ $D(0,-1.273)=67.8964 \gt 0$ and $f_{xx}=2 \gt 0$ For $(x,y)=(0,1.211)$ $D(0,1.211) =54.467 \gt 0$ and $f_{xx}=2 \gt 0$ Thus, when $D(p,q)=f_{xx}(p,q)f_{yy}(p,q)-[f_{xy}(p,q)]^2 \gt 0$ and $f_{xx}(p,q)\gt 0$ , then $f(p,q)$ is a local minimum. For $(x,y)=(\pm 0.720,0.259)$ $D(0,0.347) =-8.8550 \lt 0$ ; saddle points. Hence, Minimum: $f(0,1.211) \approx -1.403$, $f(0,-1.273) \approx -3.890$ Saddle points at $(0,0.347)$ Lowest points: $(0,-1.273, -3.890)$