相對極大與相對極小
Author: 藍邦偉老師

$ f(x)= $

1

$ 相對極大值與相對極小值 $

$x=%v$

r = 0.7 (slider from 0.1 to 1, step 0.1)
List_1 = Sequence(Segment[(-15, k), (13, k)], k, -10, 15)
J = Point(List_1)
R = (0.9631725027811541, -0.43469682987247055)
a_7 = Ray(R, J)
c_2 = Circle(R, r)
K = Intersect(c_2, a_7)
f(x) = x^(3) - 3 * x + 1
O = (8.201716469489003, 6.105145981747766)
d_a = Circle(O, 3)
A = Point(f)
a_1 = Ray(R, A)
B = Intersect(c_2, a_1)
k_1 = AffineRatio(R, B, A)
a_2 = Ray(O, O + Vector[R, A])
C = Intersect(d_a, a_2)
C' = Dilate(C, k_1, O)
C'' = If(k_1 < 1, C')
loc1 = Locus(C'', A)
D = Point(yAxis)
a_3 = Ray(R, D)
E = Intersect(c_2, a_3)
a_4 = Ray(O, Vector[R, E])
F = Intersect(d_a, a_4)
k_2 = AffineRatio(R, E, D)
F' = Dilate(F, k_2, O)
F'' = If(k_2 < 1, F')
loc2 = Locus(F'', D)
G = Point(xAxis)
a_5 = Ray(R, G)
H = Intersect(c_2, a_5)
k_3 = AffineRatio(R, H, G)
a_6 = Ray(O, Vector[R, H])
I = Intersect(d_a, a_6)
I' = Dilate(I, k_3, O)
I'' = If(k_3 < 1, I')
loc3 = Locus(I'', G)
a_8 = Ray(O, Vector[R, K])
L = Intersect(d_a, a_8)
k_4 = AffineRatio(R, K, J)
L' = Dilate(L, k_4, O)
L'' = If(k_4 < 1, L')
loc4 = Locus(L'', J)
List_2 = Sequence(Segment[(k, -7), (k, 12)], k, -13, 12)
M = Point(List_2)
a_9 = Ray(R, M)
N = Intersect(c_2, a_9)
k_5 = AffineRatio(R, N, M)
a_{10} = Ray(O, Vector[R, M])
P = Intersect(d_a, a_{10})
P' = Dilate(P, k_5, O)
P'' = If(k_5 < 1, P')
loc5 = Locus(P'', M)
c_b = Circle(O, 2.9)
Q_2: 0.3 * Corner[2] + 0.7 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
poly1, a, q_2, q_3, b = Polygon(Corner[1], Q_2, Q_3, Corner[4])
u_{y} = Vector((0, y(Q_2)), (0, y(Corner[3])))
inputBox1 = Textfield(f)
Q = Point(xAxis)
S: (x(Q), f(x(Q)))
c_a = Ray(R, S)
T = Intersect(c_2, c_a)
d = Ray(O, O + Vector[R, T])
U = Intersect(d_a, c_b, 1)
V = Intersect(d_a, c_b)
W = Intersect(d_a, c_b, 1)
Z = Intersect(d_a, c_b, 1)
A_1 = Intersect(d_a, d)
B_1 = Dilate(A_1, AffineRatio[R, T, S], O)
C_1 = If(AffineRatio[R, T, S] < 1, B_1)
loc6 = Locus(C_1, Q)
c = 1 (slider from -5 to 5, step 1)
f_1 = TaylorSeries(f, c, 3)
text1: "$ =" + (LaTeX[f_1]) + " $"
u_{x} = Vector((x(Corner[1]), 0), (x(Corner[2]), 0))
g(x) = f'(x)
h(x) = g'(x)
p(x) = h'(x)
text2: "$ c_0=f(" + (LaTeX[c]) + ") =" + (LaTeX[f(c)]) + ",c_1=f'(" + (LaTeX[c]) + ")=" + (LaTeX[g(c)]) + "$
$c_2=\frac{f''(" + (LaTeX[c]) + ")}{2!}=" + (LaTeX[FractionText[h(c) / 2]]) + ",c_3=\frac{f'''(" + (LaTeX[c]) + ")}{3!}=" + (LaTeX[FractionText[p(c) / 6]]) + "$"
text3: "$ " + (LaTeX[If[g(c) ≟ 0 ∧ (h(c) > 0), "當x=" * c * "時f(x)有最大值" * f(c), If[g(c) ≟ 0 ∧ (h(c) < 0), "當x=" * c * "時f(x)有最小值" * f(c)]]]) + " $"