05/01/18 03:04:14
>952 (>>565)
(1) {1+sin(x)}{1+cos(x)} -5sin(x)cos(x) = 1 + sin(x)+cos(x) -4sin(x)cos(x)
= 1 + 1 -2sin(2x) ≧ 2{1-sin(2x)} >0.
(2) y=tan(x) は下に凸だから、[tan(0)+tan(π/4)]/2 ≧ [tanθ + tan(π/4 -θ)]/2 ≧ tan(π/8).
また、tanθ･tan(π/4 -θ) = 1 - tanθ - tan(π/4 -θ) ≦ 1-2tan(π/8) = {tan(π/8)}^2.
∴ 1/2> A > tan(π/8) = √2 -1 > G >0.
(6) -a+b+c=x, a-b+c=y, a+b-c=z とおくと、相加･調和平均より
1/a = 2/(y+z) ≦ (1/y +1/z)/2,
1/b = 2/(z+x) ≦ (1/z +1/x)/2,
1/c = 2/(x+y) ≦ (1/x +1/y)/2.
辺々たす。
ぬるぽ