If f(x) is a continuous function such that `f(x) gt 0` for all `x gt 0` and `(f(x))^(2020)=1+int_(0)^(x) f(t) dt`, then the value of `{f(2020)}^(2019)` is equal to

Options

A. 2019
B. 2020
C. 2021
D. 2018

Answer

B

Explanation

We have,
`(f(x))^(2020)=1+underset(0)overset(x)int f(t) dt " "`....(i)
Differentiating both sides with respect to x, we get
`2020 (f(x))^(2019)f'(x)=f(x)`
`rArr 2020 (f(x))^(2018)f'(x)=1`
Integrating both sides with respect to x, we obtain
`(2020)/(2019)(f(x))^(2019)=x+C" "`....(ii)
Putting x=0 in (i) and (ii) respectively, we obtain
`{f(0)}^(2020)=1 " and "(2020)/(2019){f(0)}^(2019)=C`
`rArr f(0)=1 " and "(2020)/(2019){f(0)}^(2019)=C`
`rArr C=(2020)/(2019)` ltbr. Putting the value of C in (ii), we obtain
`(2020)/(2019){f(x)}^(2019)=x+(2020)/(2019)`
Replacing x by 2020, we obtain
`(2020)/(2019){f(2020)}^(2019)=2020+(2020)/(2019)`
`rArr {f(2020)}^(2019)=2020`