Second derivatives and local max/min

[2RIP]

If f(x,y)= xy+(1/x)+(1/y)

I find that f_x= y-(1/x²)=0 and f_y=x-(1/y²)=0.

Solving for coordinates x and y by substituting the equations, I find that y=0 or 1. However, if i try to solve for x-coordinates with y=0, I get an infinity. So does that mean I ignore the possibility of y=0?

Another question is with second-derivative test where D=f_xx*f_yy-f_xy².

I know that if D>0 and f_xx<0, then it is a local max. But what if I was told D>0 and given only f_yy<0 and was to determine if it's a local max or min. Could I make the same argument with only f_yy<0 that it is a local max?Thanks a lot

 

[HallsofIvy]

If f(x,y)= xy+(1/x)+(1/y)

I find that f_x= y-(1/x²)=0 and f_y=x-(1/y²)=0.

Solving for coordinates x and y by substituting the equations, I find that y=0 or 1. However, if i try to solve for x-coordinates with y=0, I get an infinity. So does that mean I ignore the possibility of y=0?

Yes. Any (x,y) point with y= 0 is not in the domain of the function.

Another question is with second-derivative test where D=f_xx*f_yy-f_xy².

I know that if D>0 and f_xx<0, then it is a local max. But what if I was told D>0 and given only f_yy<0 and was to determine if it's a local max or min. Could I make the same argument with only f_yy<0 that it is a local max?

-f_xy² is always negative. In order that D be positive it is necessary that f_xxf_yy be positive (and greater than f_xy²). That means that f_xx and f_yy must have the same sign. Knowing that f_yy< 0 immediately tells you that f_xx< 0.

Thanks a lot

 

[2RIP]

Oh right. Thanks for clearing that up HallsofIvy