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Let f: RvecR be a continuous function wh...

Let `f: RvecR` be a continuous function which satisfies `f(x)=` `int_0^xf(t)dtdot` Then the value of `f(1n5)` is______

Text Solution

Verified by Experts

The correct Answer is:
`(0)`
From given integral equation, F(0) = 0.
Alos, differentiating the given integral epuation w.r.t.x
`f'(x)=f(x)`
If `f(x) ne 0`
`rArr (f'(x))/f(x) = 1 rArr log f(x) = x+c`
`rArr f(x)=e^(c)e^(x)`
`therefore f(0)=0 rArr e^(c) =0,` a contradiction
`therefore f(x)=0, AA x in R`
`rArr f(ln5)=0`
Alternate Solution
Given, `f(x) = int_(0)^(x)f(t)dt `
`rArr f(0)=0 and f' (x) = f(x) `
If `f(x) ne 0`
`rArr (f'(x))/(f(x))=1 rArr ln f(x)=x+c`
`rArr f(x)=e^(c) cdot e^(x)`
`therefore f(0)=0`
`rArr e^(c)=0,` a contradiction
`therefore f(x)=0, AA x in R`
`rArr f(ln5)=0`
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