三次函數的極值
Author: 藍邦偉　老師

三次函數的極值

a_1

a_2

a_3

a_4

a=

b=

c=

d=

L=

R=

L

R

⇒極值不存在

1

x有範圍限制

1

$ 說明 $

d = 0
c = 0
b = 3
a = -1
f(x) = a * x^(3) + b * x^(2) + c * x + d
t = -4.32 (slider from x(Corner[1]) to x(Corner[2]))
A, B = Extremum(f)
u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
inputBox1 = Textfield(a)
inputBox2 = Textfield(b)
inputBox3 = Textfield(c)
inputBox4 = Textfield(d)
Delta: 4 * b^(2) - 12 * a * c
L = -1
R = 2
inputBox5 = Textfield(L)
inputBox6 = Textfield(R)
g = Function(f, L, R)
W_1: (L, f(L))
W_2: (R, f(R))
text3: "" + W_1
text12: "(" + (LaTeX[FractionText[x(W_2)]]) + "," + (LaTeX[FractionText[y(W_2)]]) + ")"
t_A: "$ (" + (LaTeX[FractionText[x(A)]]) + "," + (LaTeX[FractionText[y(A)]]) + ") $"
t_B: "$ (" + (LaTeX[FractionText[x(B)]]) + "," + (LaTeX[FractionText[y(B)]]) + ") $"
text17: "$ \Delta=b²-3ac=" + (LaTeX[b^(2)]) + "" + (LaTeX[If[a * c < 0, "+", "-"]]) + "" + (LaTeX[4 * abs(a * c)]) + "≤0 $"
e = true
q_1: f''(x(A))
q_2: f''(x(B))
q_3: f'(x(W_1) + 0.0010)
q_4: f'(x(W_2) - 0.0010)
t_f: "$f(x)=" + (LaTeX[If[abs(a) ≠ 1, FractionText[a], If[a ≟ -1, "-", ""]]]) + "x³" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1, "+x^2", If[b ≟ -1, "-x^2", If[b > 0, "+" * abs(b) * "x^2", -abs(b) * "x^2"]]]]]) + "" + (LaTeX[If[c ≟ 0, "", If[c ≟ 1, "+x", If[c ≟ -1, "-x", If[c > 0, "+" * abs(c) * "x", -abs(c) * "x"]]]]]) + "" + (LaTeX[If[d > 0, "+" * d, If[d < 0, "-" * abs(d), ""]]]) + "$
$⇒f'(x)=" + (LaTeX[If[abs(a) ≠ 1 / 3, FractionText[3 * a], If[a ≟ (-1) / 3, "-", ""]]]) + "x^2" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1 / 2, "+x", If[b ≟ (-1) / 2, "-x", If[b > 0, "+" * abs(2 * b) * "x", -abs(2 * b) * "x"]]]]]) + "" + (LaTeX[If[c > 0, "+" * c, If[c < 0, "-" * abs(c), ""]]]) + "=0$
$⇒x=" + (LaTeX[x(A)]) + ",\;" + (LaTeX[x(B)]) + "$"
t_1: "當x=" + (LaTeX[FractionText[x(A)]]) + "\;時，\;f(x)有極" + (LaTeX[If[(a > 0) ∧ (q_1 < 0), "大", If[(a > 0) ∧ (q_1 > 0), "小", If[(a < 0) ∧ (q_1 > 0), "小", If[(a < 0) ∧ (q_1 < 0), "大"]]]]]) + "值" + (LaTeX[FractionText[y(A)]]) + ""
C = (3.1401322314049533, 5.695960931630356)
t_2: "當x=" + (LaTeX[FractionText[x(B)]]) + "\;時，\;f(x)有極" + (LaTeX[If[(a > 0) ∧ (q_2 < 0), "大", If[(a > 0) ∧ (q_2 > 0), "小", If[(a < 0) ∧ (q_2 > 0), "小", If[(a < 0) ∧ (q_2 < 0), "大"]]]]]) + "值" + (LaTeX[FractionText[y(B)]]) + ""
t_3: "當x=" + (LaTeX[FractionText[x(W_1)]]) + "\;時，\;f(x)有極" + (LaTeX[If[q_3 > 0, "小", "大"]]) + "值" + (LaTeX[FractionText[y(W_1)]]) + ""
t_4: "當x=" + (LaTeX[FractionText[x(W_2)]]) + "\;時，\;f(x)有極" + (LaTeX[If[q_4 < 0, "小", "大"]]) + "值" + (LaTeX[FractionText[y(W_2)]]) + ""
h = true
Q_2: 0.2 * Corner[2] + 0.8 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
poly1, i, q_5, q_6, j = Polygon(Corner[1], Q_2, Q_3, Corner[4])