113 中山女中

[thepiano]

請參考附件

[thepiano]

第 7 題
令 f(x) = ka^x

f(x + y) = 2f(x)f(y)
ka^(x + y) = 2 * ka^x * ka^y
k = 1/2

f(x) = (1/2)a^x
f'(x) = (lna/2)a^x
f'(0) = 2，lna = 4

f''(x)/f(x) = [(1/2)(lna)^2 * a^x]/[(1/2)a^x] = 16

[thepiano]

證明第 1 題
[(a + b)/a](sinx)^4 + [(a + b)/b](cosx)^4 = 1
(b/a)(sinx)^4 + (a/b)(cosx)^4 + (sinx)^4 + (cosx)^4 = 1
(b/a)(sinx)^4 + (a/b)(cosx)^4 - 2(sinx)^2(cox)^2 = 0
[√(b/a)(sinx)^2 - √(a/b)(cosx)^2]^2 = 0
√(b/a)(sinx)^2 = √(a/b)(cosx)^2 = √(a/b)[1 - (sinx)^2]
(sinx)^2 = a/(a + b)，(cosx)^2 = b/(a + b)
代入欲證明之式子即可證出