IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

somilmiglani

find all continuous functions f:[0,1]toR differentiable in(0,1) and satisfy f(0)=f(1)=1 and 2003f'(x)+2004f(x)greater than2004

amar.gupta

2003f'(x)+2004f(x) >= 2004

hence f'(x) >= (2004/2003)*(1-f(x))

now case 1) 1-f(x) > 0 or f(x) <1

f'(x)/ [1-f(x)} >= (2004/2003)

integrate on both sides:

you will get :

-ln[1-f(x)] > = (2004/2003)x

or ln[1-f(x)] < = (2004/2003)x

or f(x) > =1+ e^ (2004/2003)x

hence f(x) >1 which is not possible as we take f(x) < 1

so no such f(x) possible in this case

case (2) f(x) > 1 or [1 - f(x)] < 0

f'(x)/ [1-f(x)] <= (2004/2003)

integrate on both sides:

you will get :

-ln[1-f(x)] < = (2004/2003)x

or ln[1-f(x)] > = (2004/2003)x

or f(x) < =1+ e^ (2004/2003)x

hence f(x) <1 which is not possible as we take f(x) > 1

so no such f(x) possible in this case

so only function possible is : f(x) =1