Question

设函数f(x)在区间[a,b]上连续,g(X)在[a,b]上连续且不编号,则在[a,b]内至少存在一

Answer 1

函数f(x)在区间[a,b]上连续，所以有最大值与最小值，分别设为M,N.不妨设g(x)≥0
N≤f(x)≤M Ng(x)≤f(x)g(x)≤Mg(x)

∫[a,b] Ng(x)dx≤ ∫[a,b]f(x)g(x)dx≤ ∫[a,b]Mg(x)dx
N∫[a,b] g(x)dx≤ ∫[a,b]f(x)g(x)dx≤ M∫[a,b]g(x)dx
N≤ {∫[a,b]f(x)g(x)dx}/∫[a,b]g(x)dx≤ M,
所以存在ξ∈[a,b]，使得，f(ξ)={∫[a,b]f(x)g(x)dx}/∫[a,b]g(x)dx,
即f(ξ)∫[a,b]g(x)dx,=∫[a,b]f(x)g(x)dx