扁頭科學 asked in 科學數學 · 7 years ago

高微(integrable and mean value)

Show that if f and g are nonnegative real functions on [a, b], with f

continuous on [a, b] and g integrable on [a, b], then there exist

x_0, x_1[∈[a, b] such that

∫(a to b) f(x)g(x)dx = f(x_0)∫(x_1 to b) g(x)dx.

2 Answers

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  • Favorite Answer

    由於f和g非負,固对所有c,a≤c≤b,

    ∫(a to c) fg≥0

    設x_1=inf{c: ∫(a to c) fg >0 },即有

    ∫(a to b) fg = ∫(x_1 to b) fg

    min{f(t): x_1≤t≤b }∫(x_1 to b) fg

    ≤ ∫(x_1 to b) fg

    ≤ max{f(t): x_1≤t≤b }∫(x_1 to b) fg

    由連續函数的中值定理,存在x_0使得

    ∫(x_1 to b) fg=f(x_0)∫(x_1 to b) g, x_1≤x_0≤b

    得証。

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  • 7 years ago

    取 x_1 = a, 用中間值定理於 f. (設 g 的定積分大於 0)

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