REPOST: There are two isolated points?

Is this some kind of mathematical riddle?

If you organize the terms a bit, you see:

f(x,y) = 5(3x² + 3xy + y²) - 6(5x⁴ + 10x³y + 10x²y² + 5xy³ + y⁴)

and those coefficients are pretty familiar:

= ( 5[(x + y)³ - x³] - 6[(x + y)⁵ - x⁵] )/y

**cheating**

I plotted:

z = g(x,y) = f(x,y) - 25/24

in 3D on Grapher (MacOS X), and it looks like the zeros occur along the y-axis, so setting x=0:

f(0,y) = 25/24

144y⁴ - 120y² + 25 = 0

(12y² - 5)² = 0

y = ±√[5/12] = ±½√[5/3] = ±0.64549722436790...

(x,y) = (0, ±√[5/12])

EDIT:

@ Scythian: You're right; they're not isolated.

On the 3D plot, the y-axis is tangent to the surface at those two points, but they are not isolated when you slice the surface with the xy-plane.

A wider look at that plot shows a pair of points (symmetrically-placed, because the whole thing is symmetric under (x,y)↔(-x,-y)) somewhere around

±(0.6, -1.3)

which appear to be local maxima with z=0, which will make them isolated as points in the xy-plane.

OK, set these to 0 (to find where ∇g = ∇f = 0) and solve:

∂f/∂x = 15(2x + y) - 30(4x³ + 6x²y + 4xy² + y³) = (15/y)((x + y)² - x² - 2(x + y)⁴ + 2x⁴) = 0

∂f/∂y = 5(3x + 2y) - 6(10x³ + 20x²y + 15xy² + 4y³) = 0

Among local maxima, besides the target pair of points, there is another pair of symmetrically-placed points

(±½,0,⅚)

and the single point (0,0,-25/24) . . [ these are for z = g = f - 25/24, not f ]

But of course, we're looking for (x,y,g(x,y)) = (a,b,0) at which ∇g = ∇f = 0

Yeah, I'm on the right track, but there's this huge boulder on it, bigger than my whole locomotive...

More later . . .

EDIT2:

@ Al: Not so fast. I didn't say what Scythian says I said. But I think he's right, so I need some time to check it.

@ Scythian: I hadn't arrived at that, but it looks like you're right:

Define α = √[5/12]; so that α² = 5/12

Then you're saying that the isolated points are

(x, y, g(x,y)) = ±(-α, 2α, 0) . . [these aren't among the points I had yet identified!]

Check:

x+y = α, so if n=2m+1 is odd, [(x+y)^n - x^n]/y = 2α^n / 2α = α^(n-1) = ([5/12]^m)

And if n=2m is even, [(x+y)^n - x^n]/y = [(α²)^m - (α²)^m]/2α = 0

f(-α,2α) = ( 5[(x + y)³ - x³] - 6[(x + y)⁵ - x⁵] )/y = 5(5/12) - 6(5/12)² = (5/12)(5 - 5/2) = 25/24

g(-α,2α) = f(-α,2α) - 25/24 = 0

∂f/∂x = 15[(x + y)² - x² - 2(x + y)⁴ + 2x⁴]/y = 0

∂f/∂y = 5(3x + 2y) - 6(10x³ + 20x²y + 15xy² + 4y³)

= 5α - 6α³(-10 + 20•2 - 15•4 + 4•8)

= 5α - 12α³ = α(5 - 12(5/12)) = 0

EDIT3:

@ Al: Scythian's answer is right; all I did was check it. It would have taken me forever to derive that solution on my own.

Give him the BA!

Fred, I don't think those points are isolated.

Edit: Fred, those were the isolated points after all. My bad! AI P, better give the man the BA.

{ (1/2)√(5/3), -√(5/3) }

{ -(1/2)√(5/3), √(5/3) }