```Date: Sep 12, 2013 3:03 AM
Author: quasi
Subject: Re: Building an Equation to find (Maximum Y) ie Highest Point on a curve!

mervynmccrabbe@gmail.com wrote:>>x^4 + y^4 + a(x^2) - a(y^2) + 2(x^2)(y^2) - bxy + c = 0>>If I have correctly evaluated >>dy/dx = 4x^3 + 2ax  + 4x(y^2) - byNo, that's not correct.The correct result is:dy/dx =    (4x^3 + 2ax + 4x(y^2) - by)/(-4y^3 + 2ay - 4y(x^2) + bx)However, it _is_ true that dy/dx = 0 implies   4x^3 + 2ax + 4x(y^2) - by = 0>If I could eliminate X and get a generalised Y-Only equation >then I could manage the rest.Ok, you asked for it ...The system of equations   x^4 + y^4 + a(x^2) - a(y^2) + 2(x^2)(y^2) - bxy + c = 0   4x^3 + 2ax + 4x(y^2) - by = 0yields the y-only equation   (d_8)(y^8)    + (d_6)(y^6)   + (d_4)(y^4)    + (d_2)(y^2)   + (d_0)   = 0where   d_8 = 256b^2          + 1024a^2   d_6 = -192(b^2)a          - 1024ac         - 768(a^3)   d_4 = -240(a^4)          - 168(a^2)(b^2)         + 1920(a^2)c         - 27(b^4)         + 288(b^2)c          + 256(c^2)   d_2 = -16(a^5)         - 4(a^3)(b^2)         + 384(a^3)c         - 1280a(c^2)         + 144a(b^2)c   d_0 = 256(c^3)         + 16(a^4)c         - 128(a^2)(c^2)quasi
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