IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

jic

Suppose that f is a function defined on the real numbers for which f(a+b) = f(a) + f(b) + ab
for all a,b. What can f be?

RyuAmakusa

f(a+b)=f(a)+f(b)+ab

f(0)=f(0)+f(0)+0 =>f(0)=0

f(0)=f(1+(-1))=f(1) + f(-1) -1=0 =>f(1) + f(-1) =1

similarly f(2) + f(-2) =4

and f(a) + f(-a) =a^2

if f(x) is odd f(a) = - f(-a) =>4=0

therefore f(x) is not odd

if f(x) is even then f(a) = f(-a)

=>2f(a) = a^2

therefore f(a)=(a^2)/2

we can see that f(x)= X²/2 satisfies the given condition ( I donot mean to say that functions are either even or odd but you get the answer )

konichiwa2x

edited.

computer001

my ans is f(x)=f'(0)*x + x^2/2
method...its not std but here goes
the imp point is tht the 2 variables a and b are independent of each other...i.e with respect to 1 the other is a constant

so d(a)/db=0
diff the given eqn wrt b
v get:
f'(a+b)=f'(b)+a
now put b=0
v get f'(a)=f'(0) + a
integrate
f(a)=a*f'(0) + (a^2)/2 + c(c=0 easy to show)
f(a)=a*f'(0) + (a^2)/2
the function cud be known exactly if f'(0) is known but i dun think it is

RyuAmakusa

koni then what's with my solution, I mean the function I got was

f(x)=x²/2 if you substitute it in your sol. you get

(x+1)² /2= x²/2 + x

=> x² + 2x + 1 = x² + 2x

then what's wrong???

you also have said that f(1)=1 but by my sol.

f(1) =1/ 2 ?? And how did you get the value of f(1)?

computer001

ryu look @ my soln...without knowing the value of f'(0) u cant generalise even in mine u get f(a)+f(-a)=0

RyuAmakusa

computer001 i think this could be the mistake

in the expression f'(a)=f'(0) + a . f'(a) is d(f(a))/db

so when you integerate you get

f(a)=b*f'(0) + ab + c

and not f(a)=a*f'(0) + (a^2)/2 + c

konichiwa2x

yeah you are right ryu, i made a calculion mistake.

According to my work,

which matches with your answer.

computer001

ryu..plz mate u r integratin wrt a..so wat ur tellin is abs impossible

RyuAmakusa

what I ment was you cannot integetate wrt a for example if you have

d(f(a))/dx= a

then you write it as d(f(x)) = a dx and integerate but can you show me now how to integerate wrt a

and koni do you have anything to say about this.

computer001

im sorry bud, but my math prof accepts im rite n unless u r above all tht..oh well neway u dun need 2 agree..those were my thots on it thts all..
and ur proof is not rigorous ..
conside soln as f(x)=x^2/2 + cx or sumtin
even then f(x) + f(-x)=x^2

computer001

u r assumin c=0..
in my soln f(0) is tht 'c' in my prv post
so when f(0)=0,i get same as urs..but else the soln will b diff

computer001

n in ur proof what abt functions tht r neither even nor odd???
u aint accountin 4 them

n if u notice my proof takes into account ur ans as well

RyuAmakusa

the question asked us to find what can be the function so i did not mind about the other functions and I did make a mention of that

Ask & Discuss Questions with Community & Experts