三次函數的根
Author: 藍邦偉  老師

三次函數根的性質

f(x)=k有一實根兩虛根

$ f(x)=k有兩重根與另一實根 $

$ f(x)=k有三相異實根 $

$ f(x)=k有一實根與兩虛根 $

f(x)=k有兩重根與另一實根

f(x)=k有三相異實根

f(x)=k一實根兩虛根

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y_p = -3.04067
x_p = 2.18473
y_o = -2
x_o = 2
y_n = 1
x_n = 1
y_m = 4
x_m = -1
y_l = -2.47346
x_l = -1.15552
y_k = 3
x_k = 2
y_j = 1
x_j = 1
y_i = -2
x_i = -1
y_h = 2.33735
x_h = -1.07149
y_g = 2
x_g = -1
y_f = -1
x_f = 1
y_e = -4
x_e = 3
y_d = 1.64409
x_d = -0.08412
y_c = 2
x_c = 0
y_b = -1
x_b = -2
y_a = -4
x_a = -4
y_8 = 2.47177
x_8 = -2.5
y_7 = -2
x_7 = -1
y_6 = 0
x_6 = 2
y_5 = 2
x_5 = 4
y_4 = -3
x_4 = 2
y_3 = -1
x_3 = 1
y_2 = -4
x_2 = -1
y_1 = -2
x_1 = 3
u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.7 * Corner[1] + 0.3 * Corner[2]
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
n = 1 (slider from 1 to 6, step 1)
A_1: (x_1, y_1)
A_2: (x_2, y_2)
A_4: (x_4, y_4)
A_3: (x_3, y_3)
f_1 = Polynomial({A_1, A_2, A_3, A_4})
B_1: (x_5, y_5)
B_2: (x_6, y_6)
B_4: (x_8, y_8)
B_3: (x_7, y_7)
f_2 = Polynomial({B_1, B_2, B_3, B_4})
E_1: (x_i, y_i)
E_2: (x_j, y_j)
E_4: (x_l, y_l)
E_3: (x_k, y_k)
f_5 = Polynomial({E_1, E_2, E_3, E_4})
D_1: (x_e, y_e)
D_2: (x_f, y_f)
D_4: (x_h, y_h)
D_3: (x_g, y_g)
f_4 = Polynomial({D_1, D_2, D_3, D_4})
C_1: (x_a, y_a)
C_2: (x_b, y_b)
C_4: (x_d, y_d)
C_3: (x_c, y_c)
f_3 = Polynomial({C_1, C_2, C_3, C_4})
F_1: (x_m, y_m)
F_2: (x_n, y_n)
F_4: (x_p, y_p)
F_3: (x_o, y_o)
f_6 = Polynomial({F_1, F_2, F_3, F_4})
Det: "" + (LaTeX[If[n ≟ 1, "a>0,D>0", If[n ≟ 2, "a<0,D>0", If[n ≟ 3, "a>0,D=0", If[n ≟ 4, "a<0,D=0", If[n ≟ 5, "a>0,D<0", "a<0,D<0"]]]]]]) + ""
k = 4.6257488675443446 (slider from y(Corner[1]) to y(Corner[3]), step 0.1)
l: y = k
M_1, m_1 = Extremum(f_1)
R = (-10.65109347744405, 2.484403510426013)
m_2, M_2 = Extremum(f_2)