Since, 1900< f(1990)< 2000
⇒ 190090<f(1990)90<200090⇒21.1<f(1990)90<22.2
∴ [f(1990)90]=21,22 (i)
Given, x−f(x)=19[x19]−90[f(x)90]
Now, there are two cases which arise.
Case I: [f(1990)90]=21
We have, 1990−f(1990)=19[199019]−90[199090]
1990-f(1990)=19(104)-90(21)
⇒ f(1990)=1904 (ii)
Case II: [f(1990)90]=22
We have, 1990−f(1990)=19[199019]−90[199090]
⇒ f(1990)=1994 (iii)
From Eqs. (ii) and (iii), we have f(1990)=1904 or 1994.