1. ## relative extrema

Hello,
Locate all relative extrema and saddle points f(x,y)=6xy - x^6 - y^6.

2. Hello, m777!

I assume you know the procedure.
. . Exactly where is your difficulty?

Locate all relative extrema and saddle points: .f(x,y) .= .6xy - x^6 - y^6

Solve the system: .f
x = 0, fy = 0

. . f
x .= .6y - 6x^5 .= .0 . . y .= .x^5 . [1]
. . f
y .= .6x - 6y^5 .= .0 . . x .= .y^5 . [2]

Substitute [1] into [2]: .x .= .(x^
5)^5 . . x .= .x^25

. . x^
25 - x .= .0 . . x(x^24 - 1) .= .0 . . x .= .0, ±1

Then: .y .= .0, ±1

. . The critical points are: .(0,0,0), (1,1,4), (-1,-1,4)

Second Partials Test: . f
xx = -30x^4 . . fyy = -30y^4 . . fxy = 6

. . Then: .D .= .(-30x^
4)(-30y^4) - 6² .= .900·x^4·y^4 - 36

At (0,0,0): .D .= .900·0²·0² - 36 .= .-6 . . . negative: saddle point at (0,0,0)

At (1,1,4): .D .= .900·1²·1² - 36 .= .+864 . . . positive: extreme point
. . f
xx .= .-30·1^4 .= .-30 . . . negative: maximum at (1,1,4)

At (-1,-1,4): .D .= .900·(-1)^
4·(-1)^4 - 36 .= .+864 . . . positive: extreme point
. . f
xx .= .-30(-1)^4 .= .-30 . . . negative: maximum at (-1,-1,4)