作者LuisSantos (但愿真的能够实现愿望)
看板trans_math
标题Re: [微分] 100 逢甲(春) 填充第2题
时间Tue Mar 4 17:48:52 2014
※ 引述《EggAche (蛋疼)》之铭言：
: 3 2 3
: If f(t)=t -3t +2, then the local minimum of g(x)=-2f((x+1) +1) happens
: at x = _________?
f(x) = x^3 - (3)(x^2) + 2
f((x+1)^3 + 1)
= ((x+1)^3 + 1)^3 - (3)(((x+1)^3 + 1)^2) + 2
= (x+1)^9 + (3)((x+1)^6) + (3)((x+1)^3) + 1
- (3)((x+1)^6 + (2)((x+1)^3) + 1) + 2
= (x+1)^9 - (3)((x+1)^3)
g(x) = (-2)((f((x+1)^3 +1)) = (-2)((x+1)^9) + (6)((x+1)^3)
g'(x) = (-18)((x+1)^8) + (18)((x+1)^2)
g"(x) = (-144)((x+1)^7) + (36)(x+1)
令 g'(x) = 0
则 (x+1)^8 - (x+1)^2 = 0
=> ((x+1)^2)((x+1)^6 - 1) = 0
=> ((x+1)^2)((x+1)^2 - 1)((x+1)^4 + (x+1)^2 + 1) = 0
∵ (x+1)^4 + (x+1)^2 + 1 > 0 , 对所有实数 x
∴ ((x+1)^2)((x+1)^2 - 1) = 0
=> ((x+1)^2)[(x+1) + 1][(x+1) - 1] = 0
=> ((x+1)^2)(x+2)(x) = 0
=> x = -2 , -1 , 0
g"(-2) = (-144)(-1) + (36)(-1) = 108 > 0
g"(1) = 0
g"(0) = -144 + 36 = -108 < 0
∴ 当 x = -2 时 , g(x) 有最小值
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