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

Options

A. `((b-a))/2`
B. `((b-a)^(2))/2`
C. `(b-a)^(2)`
D. none of these

Answer

B

Explanation

`|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`