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Differential Calculus

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18 Sep 2007 15:59:17 IST

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Can't identify the function!!

None

If f(x) is a differentiable function of x on R such that f((x+y)/3)=(f(x)+f(y))/2 for all real numbers x and y. Also f'(0)=2 . Find f'(5) (Ans = 2) I tried by taking f(x)= 2x+c, and substituting this in the given equation. It gives no result. What can another method be ?

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Neeraj Agarwal

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18 Sep 2007 17:29:43 IST

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1/3 f' ((x+y)/3) = 1/2 f'(x) ...taking derivative w.r.t. x and taking y constt.

put x=0 and y=3x

1/3 f'(x)=1/2f'(0)

since f'(0)=2,

f'(x) = 3
umm....i think i m going wrong somewhere coz i m getting f'(x) as constt. =3 but its given that f'(0) =2

joy francis

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18 Sep 2007 18:32:51 IST

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Im not sure about the partial differentiation step given by neeraj_agarwal_1990. I don't think we can take y and const...not sure though.

But still the question seems incorrect.

see,

f ' (x) = _{lim y-->0}(f (x+y) - f(x)) / y

= _{lim y-->0} ((f(3x+3y)/3) - f(x)) / y

= _{lim y-->0} ( (f(3x) + f(3y)/2) - (f(x)) /y

=_{lim y-->0} (f(3x) + f(3y) - 2f(x)) / 2y

f((x+y)/3)=(f(x)+f(y))/2

if y = 0

f (3x ) - 2f(x) = -f(0)

.: f'(x) becomes 0 for all x and all y, but it is given to be 2 at x=0?

shashidhar sundareisan

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18 Sep 2007 21:10:22 IST

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joy im not gettin f'(x) = 0.
i m gettin f'(x) = lim {y tends to 0} [f(3y) - f(0)]/2y.
which independent of x and hence constant(=2,as f'(0) = 2).
hence f'(5)= 2.
good question.

shashidhar sundareisan

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18 Sep 2007 21:16:38 IST

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but as rg 123 pointed out i m not getting f(x)=2x + c as a solution of the given functional equation.
may b experts will help.

Abhishek Ray

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18 Sep 2007 22:05:23 IST

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you are not getin f(x)=2x+c as a solution of the given functional equation because it doesnt satisfy the primary condition f((x+y)/3)=(f(x)+f(y))/2

Neeraj Agarwal

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19 Sep 2007 17:47:54 IST

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my method is ABSOLUTELY right.....

but i still think the question is wrong....lets see wat the experts say...

Priyesh

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19 Sep 2007 19:42:14 IST

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hey neeraj when u have partial differentiated taking y as constant (i.e independent of x)then how can u put y = 3x

Neeraj Agarwal

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19 Sep 2007 21:12:22 IST

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well....i know the explanation but i don't remember it....
but what i did is right and gives answers to all such questions....

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