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Given that f satisfies |f(u)-f(v)|lt=|u-...

Given that `f` satisfies `|f(u)-f(v)|lt=|u-v|forua n dv` in `[a , b]dot` Then `|int_a^bf(x)dx-b(b-a)f(a)|lt=` `((b-a))/2` (b) `((b-a)^2)/2` `(b-a)^2` (d) none of these

A

`((b-a))/2`

B

`((b-a)^(2))/2`

C

`(b-a)^(2)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`|int_(a)^(b)f(x)dx-(b-a)f(a)|=|int_(a)^(b)f(x)dx-int_(1)^(b)f(a)dx|`
`=|int_(a)^(b)(f(x)-f(a))dx|`
`le int_(a)^(b)|f(x)-f(a)|dx`
`le int_(a)^(b)|x-a|dx`
`=int_(a)^(b)(x-a)dx=((b-a)^(2))/2`
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