imbeking
幼苗
共回答了17个问题采纳率:88.2% 举报
f(x) = [x^2 + 2 + (1/x)^2 - 2] - 3(x + 1/x) - 2
= [x^2 + 2 * x * (1/x) + (1/x)^2] - 3(x + 1/x) - 4
= (x + 1/x)^2 - 3(x + 1/x) - 4
= (x + 1/x)^2 - 2 * (3/2) * (x + 1/x) + (3/2)^2 - 4 - (3/2)^2
= (x + 1/x - 3/2)^2 - 4 - (9/4)
= (x + 1/x - 3/2)^2 - (25/4)
从上式可以看出,只有当 x + 1/x - 3/2 取最小值时,函数才有最小值。
因为:
x + 1/x ≥ 2√(x * 1/x) = 2
所以,函数 f(x) 的最小值 = (2 - 3/2)^2 - (25/4) = 1/4 - (25/4) = -6
1年前
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