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

d = 0
c = 0
b = 0
a = 0
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)
text1: "f(x)=" + (LaTeX[a]) + "x³" + (LaTeX[If[b < 0, "-", "+"]]) + "" + (LaTeX[abs(b)]) + "x²" + (LaTeX[If[c < 0, "-", "+"]]) + "" + (LaTeX[abs(c)]) + "x" + (LaTeX[If[d < 0, "-", "+"]]) + "" + (LaTeX[abs(d)]) + ""
text9: "⇒f'(x)=" + (LaTeX[3 * a]) + "x²" + (LaTeX[If[b < 0, "-", "+"]]) + "" + (LaTeX[2 * abs(b)]) + "x" + (LaTeX[If[c < 0, "-", "+"]]) + "" + (LaTeX[abs(c)]) + "=0"
Delta: 4 * b^(2) - 12 * a * c
L = -30
R = 30
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)]]) + ")"
text14: "$ (" + (LaTeX[FractionText[x(A)]]) + "," + (LaTeX[FractionText[y(A)]]) + ") $"
text16: "$ (" + (LaTeX[FractionText[x(B)]]) + "," + (LaTeX[FractionText[y(B)]]) + ") $"
text11: "⇒x=" + (LaTeX[FractionText[x(A)]]) + "," + (LaTeX[FractionText[x(B)]]) + ""
text17: "$ \Delta=b²-3ac=" + (LaTeX[b^(2)]) + "" + (LaTeX[If[a * c < 0, "+", "-"]]) + "" + (LaTeX[4 * abs(a * c)]) + "≤0 $"
e = false
q_1: f''(x(A))
q_2: f''(x(B))
text22: "當x=" + (LaTeX[FractionText[x(A)]]) + "時，f(x)有極" + (LaTeX[If[q_1 < 0, "大", "小"]]) + "值" + (LaTeX[FractionText[y(A)]]) + ""
q_3: f'(x(W_1) + 0.0010)
q_4: f'(x(W_2) - 0.0010)
text23: "當x=" + (LaTeX[FractionText[x(B)]]) + "時，f(x)有極" + (LaTeX[If[q_2 > 0, "小", "大"]]) + "值" + (LaTeX[FractionText[y(B)]]) + ""
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)]]) + ""
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)]]) + ""