find critical points of optimization.?

Question

find the critical points of optimization and whether the function is a maximum or minimum at these points.

5x^2 + 4xy + 3y^2 - 52x - 56y + 13

dz/dx = 10x + 0 + 0 - 52 - 0 + 0 = 0
10x - 52 = 0
x = 5,2

dz/dy = 0 + 0 + 6y - 0 - 56 + 0 = 0
6y - 56 = 0
y = 9,34

so, theres just one critical point then? P(5,2/9,34)?

or would it be:

(-5,2/9,34)
(5,2/-9,34)
(5,2/9,34)
(-5,2/-9,34)?

i assume its just P(5,2/9,34) ..so what I did was:

d^2z/dx^2 = 10

Zxx(5,2/9,34) = 5,2*10 = 52 > 0 Minimum

d^2z/dy^2 = 6

Zyy(5,2/9,34) = 6*9,34 = 56,04 > 0 Minimum

did i do this correctly? any help/comments will be appreciated. thanks:)

Answer 1

dz/dx = 10x+4y-52=0
dz/dy = 4x+6y-56=0
30x+12y-156=0
8x+12y-112=0
22x-44=0 x= 2 and y = 8
d2z/dx^2= 10
d2z/dy^2= 6
d2z/dxdy =4
10*6-4^2>0 and d2z/dx^2 >0 locxal minimum