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Let f:RtoR be given f(x)=(x-1)(x-2)(x-5)...

Let `f:RtoR` be given `f(x)=(x-1)(x-2)(x-5)` ltBrgt Define `F(x)=underset(0)overset(x)f(t)dt,x gt 0` the following options is/are correct?

A

`F(x)ne0, AA x in (0,5)`

B

F(x) has two local maxima and one local minima in `(0, infty)`

C

F(x) has a local maxima at x=2

D

F(x) has a local minima at x=1

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
`f(x)=(x-1)(x-2)(x-5)`
`F(x)=underset(0)overset(x)intf(t)dt` as F(1) is negative and F(2) is also negative so F(x) cannot be zero of `x in (0,5)`. So option (A) is orrect

`thereforeF'(x)=f(x)=(x-1)(x-2)(x-5)`
`impliesF(x)` has two local minima points at `x=1` and x=5
`F(x)` has one local minima point at x=2
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