Let `f(x)` be a function defined by `f(x)=int_(1)^(x)t(t^(2)-3t+2)dt,1ltxlt3` then the maximum value of `f(x)` is

Answer

2

Explanation

`f^(')(x)=x(x^(2)-2x+2)=x(x-1)(x-2)` the sign scheme for `f^(')(x)` is as below

`becausef^(')(x)lt0` in `1lexle2` and `f^(')(x)ge0` in `2lexle5`
`becausef(x)` is decreasing in {1,2} and increasing in [2,3]
maximum `f(x)=` the greatest among `{f(1),f(3)}`
`f(1)=int_(1)^(1)x(x^(2)-3x+2)dx=0,f(3)=int_(1)^(3)x(x^(2)-3x+2)dx=2`
`because` maximum `f(x)=2`