高等数学吧 关注：471,954贴子：3,875,504
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证明：
令 c = (1/n) * (1, n ) ∑ Xi , 在 x = c 处对 f(x) 进行泰勒展开，有
f(x) = f(c) + f '(c) (x- c) + [ f "(ξ) / 2! ] * (x - c)^2 ≥ f(c) + f '(c) (x- c)
f(Xi) ≤ f(c) + f '(c) (Xi - c )
(1, n ) ∑ f(Xi) ≥ (1, n ) ∑ f(c) + (1, n ) ∑ f '(c) (Xi - c )
(1, n ) ∑ f(Xi) ≥ (1, n ) ∑ f(c) = n * f(c)
f [ (1/n) * (1, n ) ∑ Xi ] ≤ (1/n) * (1, n ) ∑ f(Xi)
令 c = (1/n) * (1, n ) ∑ Xi , 在 x = c 处对 f(x) 进行泰勒展开，有
f(x) = f(c) + f '(c) (x- c) + [ f "(ξ) / 2! ] * (x - c)^2 ≥ f(c) + f '(c) (x- c)
f(Xi) ≤ f(c) + f '(c) (Xi - c )
(1, n ) ∑ f(Xi) ≥ (1, n ) ∑ f(c) + (1, n ) ∑ f '(c) (Xi - c )
(1, n ) ∑ f(Xi) ≥ (1, n ) ∑ f(c) = n * f(c)
f [ (1/n) * (1, n ) ∑ Xi ] ≤ (1/n) * (1, n ) ∑ f(Xi)
