设f(x)在[a,b]上连续,在(a,b)内可导且f′(x)≤0,并有 证明:在(a,b)内有F'(x)≤0
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设f(x)在[a,b]上连续,在(a,b)内可导且f′(x)≤0,并有 证明:在(a,b)内有F'(x)≤0
设f(x)在[a,b]上连续,在(a,b)内可导且f′(x)≤0,并有 证明:在(a,b)内有F'(x)≤0
设f(x)在[a,b]上连续,在(a,b)内可导且f′(x)≤0,并有 证明:在(a,b)内有F'(x)≤0
F(x)=∫ [a-->x] f(t)dt/(x-a)
F'(x)=( f(x)(x-a)-∫ [a-->x] f(t)dt )/(x-a)^2
由积分中值定理,存在ξ∈(a,x),使∫ [a-->x] f(t)dt=f(ξ)(x-a)
则F'(x)=( f(x)(x-a)-f(ξ)(x-a) )/(x-a)^2
=(f(x)-f(ξ))/(x-a)
由x在(a,b)内,x>a,由ξ∈(a,x),则ξ
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