Golden Ratio in Inflection Points of Quartic Curve

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

negative 10

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

negative 10

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

negative 10

Expression 27:

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

Expression 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

equals

negative 13

Expression 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

equals

Expression 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

equals

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

equals

negative 96

Expression 33:

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

Expression 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

equals

0.9 6 3 2 6 2 8 7 7 6 6 5

Expression 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

equals

5.5 3 6 7 3 7 1 2 2 3 4

Expression 37:

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

Expression 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

equals

9.3 7 5

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

equals

negative 6.5

Expression 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

equals

negative 15.5 8 3 3 3 3 3 3 3 3

Expression 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.

Expression 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

equals

negative 1.8 6 3 2 9 9 6 5 2 2 1

Expression 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

equals

8.3 6 3 2 9 9 6 5 2 2 1

Expression 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.

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

equals

2.2 8 1 6 4 2 0 2 1 4 6