09/06/21 02:29:17
>>6 (上), >>8
n=2m (偶数)のとき cos(x)^2 - (1/2) = ξ, とおく。|ξ| ≦ 1/2, 与式は
(与式) = (1/2 - ξ)^m + (1/2 + ξ)^m = 2∑[k=0,[m/2]] Ｃ[m,2k] (1/2)^(m-2k) ξ^(2m),
ξ=±1/2 のとき最大値 1,
ξ=0 のとき最小値 (1/2)^(m-1),
nが奇数のとき、与式を g(x) とおくと,
g '(x) = n･sin(x)cos(x){sin(x)^(n-2) - cos(x)^(n-2)},
最大値 g(0) = g(π/2) = 1,
最小値 g(π) = g(3π/2) = -1,
なお、
極小値 g(π/2) = (1/2)^(n/2 -1),
極大値 g(3π/2) = -(1/2)^(n/2 -1),