Question

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

Answer 1

i got fx(x,y)= 3x^2y+24 fy(xy)= x^3-8

Answer 2

fxx (x,y) = 6xy+ 24 fyy(x,y)= 3x^2
so x=0,-8, 2

Answer 3

What do you mean x=0, -8, 2? What is that the result of?

Answer 4

setting the fx and fy =0

Answer 5

i am not sure of what i am suppose to do really past finding the partial derv.

Answer 6

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

Answer 7

i think you meant that since you did fxx right though

Answer 8

From fx and fy, you should set each to 0 and solve as simultaneous equations and you get (x,y) critical points.

Answer 9

is fxy= 3x^2?

Answer 10

yes

Answer 11

that is where i got the values for x. those are the critical points, right?

Answer 12

Critical points are from fx and fy. Set each to 0; solve as simultaneous equations for (x,y) critical points.

Answer 13

x=0,-8,2

Answer 14

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

Answer 15

Evaluate fxx at critical points
Evaluate fxy at critical points
Evaluate fyy at critical points

Answer 16

Let's call above evaluations
A (fxx)
B (fxy)
C (fyy)

Answer 17

it looks that the only only critical number is (2,-4)

Answer 18

point*

Answer 19

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

Answer 20

let me work at this

Answer 21

its the saddle point

Answer 22

Good work