How to prove that this function is primitive recursive?

How do I prove this conjecture for a general quartic function?

• How do I prove that for every quartic function the following is true: If R and Q are the points of inflection and P and S are the points colinear to the points of inflection that reintersect the function then the ratio always is the following PQ:QR:RS = 1: 1.618:1

• Answer:

  Given the general quartic: Y = AX^4 + BX^3 + CX^2 + DX + E, divide by A and put X = x - (A/4) to obtain a simpler equation: y = x^4 + ax^2 + bx + c ...(1) That gives a quartic which has been scaled vertically and shifted horizontally, but does not change the required ratios. Differentiating twice: y' = 4x^3 + 2ax + b y'' = 12x^2 + 2a Inflection points are: x1 = - sqrt(-a/6) x2 = sqrt(-a/6). The corresponding y co-ordinates are: y1 = - 5a^2 / 36 - b sqrt(-a/6) + c y2 = - 5a^2 / 36 + b sqrt(-a/6) + c The line joining these has equation: y = (y2 - y1)x / (x2 - x1) + y1 which yields: y = bx - 5a^2 / 36 + c ...(2) Equating (1) and (2) gives: x^4 + ax^2 + 5a^2 / 36 = 0 As x1 and x2 are known roots of this equation, it can be factorised: (x^2 + a/6)(x^2 + 5a/6) = 0 The remaining roots are: x3 = -sqrt(-5a/6) x4 = sqrt(-5a/6) The ratio of the distances required is equal to the ratio of their projections on the x axis. This is equal to: (sqrt(-a/6) - [-sqrt(-a/6)] ) / ( sqrt(-5a/6) - sqrt(-a/6) ) = 2 / (sqrt(5) - 1) = (sqrt(5) + 1) / 2.