Let `f:[0,4] in R` be acontinuous function such that `|f(x)|le2 "for all" x in [0,4] and int_(0) ^(4)f(t)=2`. Then, for all ` x in [0,4]` , the value of `int_(0)^(k) f(t)` dt lies in the in the interval

Options

A. `[-6+2x,10-2x]`
B. `[-12+2x,-7+2x]`
C. `[ 11-2x,17+2x]`
D. `[-8-2x,6-2x]`

Answer

A

Explanation

Let `phi(x)=underset(0)overset(x)int f(t)` dt. Then `phi(x)` is continuou on [0.4] and is differentiable on (0,4) such `phi'(c)=(phi(4)-phi(x))/(4-x)"for some" c in (x,4)`
`rArr f(c)(2-phi(x))/(4-x)`
`rArr |(c)|=|(2-phi(x))/(4-x)|`
`rArr |f(c)|=|(phi(x)-2)/(4-x)|`
`rArr|(phi(x)-2)/(4-x)|le2`
`rArr |phi(x)-2|le8-2x`
`rArr -8+2xlephi(x)-2le8-2x`
`rArr 2x-6lephi(x)le-2x`
`rArr phi(x)=[12x-6,10-2x]`