05/01/17 21:58:51
>941
左辺 = (1/2)sin(2A) + (1/2)sin(2B) + (1/2)sin(2C) ≦ (3/2)sin(2π/3) = (3/4)√3.
>944
f(θ) = (tanθ)^n (cosθ)^2.
f '(θ) = n(tanθ)^(n-1) -2(tanθ)^(n+1)･(cosθ)^2.
f "(θ) = n(n-1)(tanθ)^(n-2)/(cosθ)^2 -2(n+1)(tanθ)^n + 4{(tanθ)^(n+2)}(cosθ)^2
= {n(n-1)[1+(tanθ)^2]^2 -2(n+1)(tanθ)^2[1+(tanθ)^2] +4(tanθ)^4}{(tanθ)^(n-2)}(cosθ)^2
= {n(n-1) +2(n^2 -2n-1)(tanθ)^2 +(n-1)(n-2)(tanθ)^4}{(tanθ)^(n-2)}(cosθ)^2
= {-D_n +[(n-1)^2 -2 +(n-1)(n-2)(tanθ)^2]^2}{(tanθ)^(n-2)}(cosθ)^2 /[(n-1)(n-2)].
ここに、判別式 D_n = [(n-1)^2 -2]^2 -n(n-2)(n-1)^2 = 4 -3(n-1)^2.
n≦1-(2/3)√3 = -0.154700538379252… または n≧1+(2/3)√3 = 2.154700538379252… のとき
D_n ≦0, f "≧0 (下に凸).
∴ (x^n)/(1+x^2) + (y^n)/(1+y^2) + (z^n)/(1+z^2) ≧ (3/4)(√3)^n.
ぬるぽ