四次函數的極值
Author: 藍邦偉　老師

四次函數的極值

a_1

a_2

a_3

a_4

a=

b=

c=

d=

L=

R=

L

R

1

x有範圍限制

e=

e

e = -2
a = -1
b = 0
c = 4
d = 0
t = -0.7276363636363625 (slider from x(Corner[1]) to x(Corner[2]))
f(x) = a * x^(4) + b * x^(3) + c * x^(2) + d * x + e
A, B, C = 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 = -2
R = 2
inputBox5 = Textfield(L)
inputBox6 = Textfield(R)
g = Function(f, L, R)
W_1: (L, f(L))
W_2: (R, f(R))
t_L: "" + W_1
t_R: "(" + (LaTeX[FractionText[x(W_2)]]) + "," + (LaTeX[FractionText[y(W_2)]]) + ")"
t_1 = 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)
text25: "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)]]) + ""
inputBox7 = Textfield(e)
q_5: f''(x(A))
text24: "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_f: "$f(x)=" + (LaTeX[If[abs(a) ≠ 1, FractionText[a], If[a ≟ -1, "-", ""]]]) + "x⁴" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1, "+x^3", If[b ≟ -1, "-x^3", If[b > 0, "+" * abs(b) * "x^3", -abs(b) * "x^3"]]]]]) + "" + (LaTeX[If[c ≟ 0, "", If[c ≟ 1, "+x^2", If[c ≟ -1, "-x^2", If[c > 0, "+" * abs(c) * "x^2", -abs(c) * "x^2"]]]]]) + "" + (LaTeX[If[d ≟ 0, "", If[d ≟ 1, "+x", If[d ≟ -1, "-x", If[d > 0, "+" * abs(d) * "x", -abs(d) * "x"]]]]]) + "" + (LaTeX[If[e > 0, "+" * e, If[e < 0, "-" * abs(e), ""]]]) + "$
$⇒f'(x)=" + (LaTeX[If[abs(a) ≠ 1 / 4, FractionText[4 * a], If[a ≟ (-1) / 4, "-", ""]]]) + "x^3" + (LaTeX[If[b ≟ 0, "", If[b ≟ 1 / 3, "+x^2", If[b ≟ (-1) / 3, "-x^2", If[b > 0, "+" * abs(3 * b) * "x^2", -abs(3 * b) * "x^2"]]]]]) + "" + (LaTeX[If[c ≟ 0, "", If[c ≟ 1 / 2, "+x", If[c ≟ (-1) / 2, "-x", If[c > 0, "+" * abs(2 * c) * "x", -abs(2 * c) * "x"]]]]]) + "" + (LaTeX[If[d > 0, "+" * d, If[d < 0, "-" * abs(d), ""]]]) + "=0$
$⇒x=" + (LaTeX[x(A)]) + "," + (LaTeX[x(B)]) + "," + (LaTeX[x(C)]) + "$"
D = (3.6194710743801664, 4.786870022539447)
text22: "當x=" + (LaTeX[x(A)]) + "時，\;f(x)有極" + (LaTeX[If[q_1 < 0, "大", "小"]]) + "值" + (LaTeX[y(A)]) + ""
text23: "當x=" + (LaTeX[x(B)]) + "時，\;f(x)有極" + (LaTeX[If[q_2 > 0, "小", "大"]]) + "值" + (LaTeX[y(B)]) + ""
text29: "當x=" + (LaTeX[x(C)]) + "時，\;f(x)有極" + (LaTeX[If[q_5 > 0, "小", "大"]]) + "值" + (LaTeX[y(C)]) + ""
t_A: "$ " + (LaTeX[A]) + "$"
t_B: "$ " + (LaTeX[B]) + " $"
t_C: "" + (LaTeX[C]) + ""