The difference between the greateset ahnd least vlaue of the function `f(x)=int_(0)^(x) (6t^2-24)" dt on " [1,3] " dt on " [1,3]` is

Options

A. 14
B. 10
C. 5
D. 4

Answer

A

Explanation

We have
`f(x)underset(0)overset(x)int (6t^2-24)dt on [1,3] ` is
`f(x)=6x2-24=6(x-2)(x+2)`
Signs of f(x) for differenet values of x are shown in

We observe that f(x) is decreasing in the interval (1,2) and inreasing in ther interval (2,3) Therefore , the least value of f(x) is given by
`f(2)=underset(0)overset(2)int (6t^2-24)dt =[2t^3-24t[_0^2`=-32
and Greatest value of f(x)=Max {f(1),(f(x)}
We have
`f(1)=underset(0)overset(1)int (6t^2-24)dt =-20,f(3)=underset(0)overset(1)int(6t^2-24)dt =-18`
`therefore` Greatest value f(x)=-18
Hence,required difference =-18-(-32)=14