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Integral Calculus

Blazing goIITian

Joined:	13 Aug 2008
Post:	1313

2 Nov 2008 20:39:04 IST

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511

Sample paper qn for JEE 2006 ....

None

A function f : R® R satisfies f(x+y) = f(x) + f(y) for all x,y Î R

and is continuous throughout the domain.

If I₁ + I₂ + I₃ + I₄+ I₅ = 450, where I_n = n. ₀òⁿ f(x) dx.

Find f(x).

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Comments (3)

Decoder

Blazing goIITian

Joined: 1 Apr 2007

Posts: 1084

2 Nov 2008 22:01:49 IST

Like 5 people liked this

look..u now by concavity-convexivity..tht given func. is a straight line func.
tht too of form y=kx...as f(0) = 0...

we don noe..k..tht only we have to calculte..

so .. 450 = k sum{ n^3 / 2}

900 = k (225)..

k = 4..

so..y =4x...

rate heavily..if correct..boink mildly if wrong .. :)

Nitish

Hot goIITian

Joined: 10 Jul 2008

Posts: 110

2 Nov 2008 22:41:17 IST

Like 6 people liked this

@Decoder

Very well Done!

.........................................................................(1)

f(x+y) = f(x) + f(y)

Put x=n and y= -x

f(n-x) = f(n) + f(-x)

Also, Putting x=y=0 we get f(0) = 0.

Thus, f(x) + f(-x) = 0

..........................................................(2)

Add (1) and (2)

.........................................................................................(3)

Now Put x=y=1

f(2) = 2f(1)

Put x=2 y=1

f(3) = f(2) + f(1)

= 2f(1) + f(1)

= 3f(1)

f(n) = nf(1)................................................................................(4)

Now,

Use (3) (4) and the above condition to get f(1) = 900/225

Thus, f(1)=4

And Since f(x) = xf(1) for all x,

f(x) = 4x

Mirka

Blazing goIITian

Joined: 13 Aug 2008

Posts: 1313

22 Mar 2009 19:30:16 IST

Like 0 people liked this

MY 1000 TH POST !!!

I am renewing this thread 4 everyone to see ....

this is a really good Qsn with equally superb solutions !!!

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