if f((x+y)/3) = (2+f(x) +f(y))/3 for all real x, y and f’ (2) =2 and f if differentiable function
Then can we find f (x) .in question nothing is said about f(x)(no further information is given)
but they are finding the answer as 2x+2
if f((x+y)/3) = (2+f(x) +f(y))/3 for all real x, y and f’ (2) =2 and f if differentiable function
Then can we find f (x) .in question nothing is said about f(x)(no further information is given)
but they are finding the answer as 2x+2
Take the partial derivative with respect to y to get
$\displaystyle \frac{1}{3}\frac{\partial f}{\partial y}\left(\frac{x+y}{3} \right)=\frac{1}{3}\frac{\partial f}{\partial y}(y) $
Now if we evaluate this at the point x=3x y=0)we get (I will now use primes to represent the derivative with respect to y)
$\displaystyle f'(x)=f'(0) $
Since the derivative is constant the function must me linear! If we plug in x=2 we get
$\displaystyle f'(2)=f'(0) =2$
So the equation must have the form
$\displaystyle f(x)=2x+b$
Can you finish from here?
this is not a very rigorous solution but is good to see.
put x=0,y=0 to get f(0)=2.
the given condition becomes
$\displaystyle f(\frac{0+x+y}{3})=\frac{f(0)+f(x)+f(y)}{3}$.....(1)
note that the centroid of the triangle $\displaystyle OAB$ with $\displaystyle O=(0,f(0)),\,A=(x,f(x)),\,B=(y,f(y))$ is $\displaystyle (\frac{0+x+y}{3},\frac{f(0)+f(x)+f(y)}{3})$.
i am assuming that the curve does not change its concavity(this is where it becomes non-rigorous but this may be established by putting y=x and then using second derivative test).
The centroid of the triangle is inside the triangle but from (1) the centroid is on the curve Y=f(X) and hence outside ot on the triangle .... to see this draw a curve which is concave downwards and mark the three points OAB(this part is also non-rigorous)
so the curve of 'f' has to be a straight line.
put f(X)=aX+b and use f'(2)=2.
i am assuming that the curve does not change its concavity(this is where it becomes non-rigorous but this may be established by putting y=x and then using second derivative test).
The centroid of the triangle is inside the triangle but from (1) the centroid is on the curve Y=f(X) and hence outside ot on the triangle .... to see this draw a curve which is concave downwards and mark the three points OAB(this part is also non-rigorous
bhaiya i cant understand even a single line .but it looks beautiful. can you explain in a very simple way whole of the above part