Extrema of Functions of two variables: Find the critical points of the function and from the form of the function determine whether a relative maximum or relative minimum occurs at each point.

f(x,y,z)=9-(x(y-1)(z+2))^2

Question

Extrema of Functions of two variables: Find the critical points of the function and from the form of the function determine whether a relative maximum or relative minimum occurs at each point.

f(x,y,z)=9-(x(y-1)(z+2))^2

anonymous · Answer

Question since there are three terms instead of two does d become the following:

$$d=f _{xx(a,b)f _{yy}}(a,b)f _{zz}(a,b)-f _{xy}(x,y)^2$$

^ this is the formula to test for relative extrema

anonymous · Answer

Is this right:

$$f _{x}(x,y,z)=-2x(y-1)^2(z+2))^2=0$$
$$f _{y}(x,y,z)=-2x^2(y-1)(z+2))^2=0$$
$$f _{x}(x,y,z)=-2x^2(y-1)^2(z+2))=0$$

I get critical points of (0,1,-2)

anonymous · Answer

The second derivatives are:

$$f _{xx}(x,y,z)=-2(y-1)^2(z+2))^2$$
$$f _{yy}(x,y,z)=-2x^2(1)(z+2))^2$$
$$f _{zz}(x,y,z)=-2x^2(y-1)^2(1))$$

anonymous · Answer

So plugging the critical points into the problem d above (considering i'm using the right equation and its not the test for extrema of three variables) then my d is 0 which makes the problem inconcusive. Is this right?