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The function f(x)=int(-1)^x t(e^t-1)(t-1...

The function `f(x)=int_(-1)^x t(e^t-1)(t-1)(t-2)^3(t-3)^5dt` has a local minimum at `x=` 0 (b) 1 (c) 2 (d) 3

A

`0`

B

`1 `

C

`2`

D

`3`

Text Solution

Verified by Experts

The correct Answer is:
B, D
`f(x) = int_(-1) ^(x) t (e ^(t) - 1) ( t - 1) ( t - 2)^(3) ( t- 3)^(5) dt `

`f ' (x) = (d)/(dx) int_(-1) ^(x) t (e^(t) - 1) ( t-1) (t - 2) ^(3) ( t- 3 ) ^(5) dt`
`" " = x (e ^(x) - 1) (x -1)(x - 2)^(3) (x - 3) ^(5) xx 1 `
`" " [ because (d)/(dx) int_(phi(x)) ^(psi(x)) f(t) dt = f { psi (x) } psi' (x) - f(phi (x) } phi' (x)]`
For local minimum, `f' (x) = 0`
`rArr " "x = 0, 1, 2 , 3`
Let ` " " f' (x) = g(x) =x ( e^(x) - 1) (x - 1) (x - 2)^(3) (x - 3 ) ^(5)`
Using sign rule,

This shows that `f(x)` has a local minimum at ` x= 1 and x = 3 ` and maximum at ` x = 2`
Therefore, (b) and (d) are the correct answers.
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