AP EAMCET202318 May 2023Evening ShiftMathematicsDifferentiationActual
If f ( x ) is a function such that f ^ ( x )= f ^2( x )-1 and f(0)=1 , then f(1)=
Options
- Ae ⁻²+1 2 e
- Be ^2+1 2 e
- Ce ^2-1 2 e
- De ⁻²-1 2 e
Correct answer
B. e ^2+1 2 e
Step-by-step solution
aligned & f ^ ( x )= f ( x ) 1 & f ^ ( x ) f ^2( x )-1 =1 aligned Integrating both sides, we get aligned & [f(x)+ f^2(x)-1 ]=x+c & At x=0, [f(0)+ f^2(0)-1 ]=0+c .....(i) & [1+ 1-1 ]=0+c 0=c aligned Putting the value of c in eq ^ n (i), we get aligned & [ f ( x )+ f ^2( x )-1 = x +0 . & At x =1: [ f (1)+ f ^2(1)-1 ]=1 & f (1)+ f ^2(1)-1 = e & f ^2(1)-1 = e - f (1) & f ^2(1)-1= e ^2+ f ^2(1)-2 ef (1) f (1)= 1+ e ^2 2 e aligned