2D 座標架
Author: 藍邦偉  老師

$ O $

$ 指數函數圖形凹向上 $

$  A$

$ B $

$M$

$ A_1(x_1,a^{x_1}) $

$ B_1(x_2,a^{x_2}) $

$ \frac{a^{x_1}+a^{x_2}}{2} $

$a^{\frac{x_1+x_2}{2}}  $

$ Pf： $
$∵\frac{a^{x_1}+a^{x_2}}{2}≥\sqrt{a^{x_1}a^{x_2}}$
$\;\;\;\;\;\;=\sqrt{a^{x_1+x_2}}=a^{\frac{x_1+x_2}{2}}$

1

$ 顯示證明 $

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: 0.75 * Corner[1] + 0.25 * 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])
f(x) = 2^(x + 0.13655176761406063) + 0.01706897095175758
A = Point(u)
B = Point(u)
A_1: (x(A), f(x(A)))
B_1: (x(B), f(x(B)))
M = Midpoint(A, B)
M_1: (x(M), f(x(M)))
a_1 = Segment(A, A_1)
a_2 = Segment(B, B_1)
a_4 = Segment(A_1, B_1)
a_a = OrthogonalLine(M, u)
M_2 = Intersect(a_a, a_4)
a_3 = Segment(M, M_2)
m = Distance(M, A)
n = Distance(M, B)
R_1: (0, y(M_1))
R_2: (0, y(M_2))
w = Vector(M_1, R_1)
z = Vector(M_2, R_2)
t_{Prp}: "$ \frac{a^{x_1}+a^{x_2}}{2}≥a^{\frac{x_1+x_2}{2}} $
$x_1=" + (LaTeX[x(A)]) + ",x_2=" + (LaTeX[x(B)]) + "$
$ \frac{x_1+x_2}{2}=" + (LaTeX[x(M)]) + "$
$a^{x_1}=" + (LaTeX[y(A_1)]) + ",a^{x_2}=" + (LaTeX[y(B_1)]) + "$
$\frac{a^{x_1}+a^{x_2}}{2}=" + (LaTeX[y(M_2)]) + "$
$a^{\frac{x_1+x_2}{2}}=" + (LaTeX[y(M_1)]) + "$"
a = false