Find the local maximum and minimum values and saddle point(s) of the function. f(x, y) = x3y + 12x2 - 8y

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Find the local maximum and minimum values and saddle point(s) of the function. f(x, y) = x3y + 12x2 - 8y

Mathematics
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i got fx(x,y)= 3x^2y+24 fy(xy)= x^3-8
fxx (x,y) = 6xy+ 24 fyy(x,y)= 3x^2 so x=0,-8, 2
What do you mean x=0, -8, 2? What is that the result of?

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setting the fx and fy =0
i am not sure of what i am suppose to do really past finding the partial derv.
fx=3x^2y+24x (this was the only mistake i seen from above so far) fy=x^3-8 fxx=6xy+24 fyy=3x^2
i think you meant that since you did fxx right though
From fx and fy, you should set each to 0 and solve as simultaneous equations and you get (x,y) critical points.
is fxy= 3x^2?
yes
that is where i got the values for x. those are the critical points, right?
Critical points are from fx and fy. Set each to 0; solve as simultaneous equations for (x,y) critical points.
x=0,-8,2
fx=0 implies 3x^2y+24x=0 3x(xy+8)=0 3x=0 => x=0 xy+8=0 => x=-8/y fy=0 implies x^3-8=0 when x=2 x=2 => 2=-8/y =>y=-4
Evaluate fxx at critical points Evaluate fxy at critical points Evaluate fyy at critical points
Let's call above evaluations A (fxx) B (fxy) C (fyy)
it looks that the only only critical number is (2,-4)
point*
If AC-(B)^2>0 and A>0 local min If AC-B^2>0 and A<0 local max If AC-B^2<0 saddle point
let me work at this
its the saddle point
Good work

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