Golden Ratio in Inflection Points of Quartic Curve

Uitdrukking 24: "c" Subscript, 1 , Baseline equals 6

Uitdrukking 25: "c" Subscript, 2 , Baseline equals 1

Uitdrukking 26: "c" Subscript, 3 , Baseline equals 8

Uitdrukking 27:

Find the coefficients of the polynomial whose roots are at x = c0, c1, c2, c3.

Uitdrukking 29: "B" Subscript, 4 , Baseline equals negative "c" Subscript, 0 , Baseline minus "c" Subscript, 1 , Baseline minus "c" Subscript, 2 , Baseline minus "c" Subscript, 3 , Baseline

Uitdrukking 30: "C" Subscript, 4 , Baseline equals "c" Subscript, 2 , Baseline "c" Subscript, 3 , Baseline plus "c" Subscript, 1 , Baseline "c" Subscript, 3 , Baseline plus "c" Subscript, 0 , Baseline "c" Subscript, 3 , Baseline plus "c" Subscript, 1 , Baseline "c" Subscript, 2 , Baseline plus "c" Subscript, 0 , Baseline "c" Subscript, 2 , Baseline plus "c" Subscript, 0 , Baseline "c" Subscript, 1 , Baseline

Uitdrukking 31: "D" Subscript, 4 , Baseline equals negative "c" Subscript, 1 , Baseline "c" Subscript, 2 , Baseline "c" Subscript, 3 , Baseline minus "c" Subscript, 0 , Baseline "c" Subscript, 2 , Baseline "c" Subscript, 3 , Baseline minus "c" Subscript, 0 , Baseline "c" Subscript, 1 , Baseline "c" Subscript, 3 , Baseline minus "c" Subscript, 0 , Baseline "c" Subscript, 1 , Baseline "c" Subscript, 2 , Baseline

Uitdrukking 32: "E" Subscript, 4 , Baseline equals "c" Subscript, 0 , Baseline "c" Subscript, 1 , Baseline "c" Subscript, 2 , Baseline "c" Subscript, 3 , Baseline

Uitdrukking 33:

Use the quadratic formula to find the x-coordinates of the two inflection points of this curve.

Uitdrukking 35: "x" Subscript, 0 , Baseline equals StartFraction, negative StartRoot, 9 "B" squared minus 24 "C" Subscript, 4 , Baseline , EndRoot minus 3 "B" Subscript, 4 , Baseline Over 12 , EndFraction

Uitdrukking 36: "x" Subscript, 1 , Baseline equals StartFraction, StartRoot, 9 "B" squared minus 24 "C" Subscript, 4 , Baseline , EndRoot minus 3 "B" Subscript, 4 , Baseline Over 12 , EndFraction

Uitdrukking 37:

Calculate intermediate constants that are used to find the rest of the information.

Uitdrukking 39: "S" Subscript, "i" , Baseline equals StartFraction, "f" left parenthesis, "x" Subscript, 1 , Baseline , right parenthesis minus "f" left parenthesis, "x" Subscript, 0 , Baseline , right parenthesis Over "x" Subscript, 1 , Baseline minus "x" Subscript, 0 , Baseline , EndFraction

Uitdrukking 40: "B" Subscript, 2 , Baseline equals "B" Subscript, 4 , Baseline plus "x" Subscript, 0 , Baseline plus "x" Subscript, 1 , Baseline

Uitdrukking 41: "C" Subscript, 2 , Baseline equals "C" Subscript, 4 , Baseline plus left parenthesis, "x" Subscript, 0 , Baseline plus "x" Subscript, 1 , Baseline , right parenthesis "B" Subscript, 4 , Baseline plus "x" Subscript, 0 , Baseline "x" Subscript, 1 , Baseline plus "x" squared plus "x" squared

Uitdrukking 42:

Use the quadratic formula again to find the x-coordinates of the other two points of intersection that the line has with the curve.

Uitdrukking 44: "x" Subscript, 2 , Baseline equals StartFraction, negative StartRoot, "B" squared minus 4 "C" Subscript, 2 , Baseline , EndRoot minus "B" Subscript, 2 , Baseline Over 2 , EndFraction

Uitdrukking 45: "x" Subscript, 3 , Baseline equals StartFraction, StartRoot, "B" squared minus 4 "C" Subscript, 2 , Baseline , EndRoot minus "B" Subscript, 2 , Baseline Over 2 , EndFraction

Uitdrukking 46:

The ratio below exists on the interval [phi^2, 5], where phi is the golden ratio. It is equal to 5 when there are only a quartic and a squared term in the polynomial. The ratio is the golden ratio squared when there are only a quartic and a cubic term in the polynomial.

Uitdrukking 48: StartFraction, "x" Subscript, 3 , Baseline Over "x" Subscript, 1 , Baseline , EndFraction squared

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