IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

hsbhatt

Ok to the list of standard functional equations, add the Jensen's equation:

$f \left(\frac{x+y}{2} \right ) = \frac{f(x)+f(y)}{2}$

with the general solution f(x) = ax+b as indicated above.

The proof, if you dont want to invoke the arguments of convexity and concavity is as follows:

Setting f(0) = a and then y = 0, we get

$f \left( \frac{x}{2} \right) = \frac{f(x)+a}{2}$

So, now

$\frac{f(x)+f(y)}{2} = f \left( \frac{x+y}{2} \right) = \frac{f(x+y)+a}{2} \\ \\<br/>\Rightarrow f(x) + f(y) = f(x+y)+a \ \text{or} \ [f(x)-a]+[f(y)-a] = [f(x+y) - a]$

Setting g(x) = f(x) - a,

we get g(x)+g(y) = g(x+y)

Now we have reduced it to the standard Cauchy functional equation with solution g(x) = mx

Hence f(x) = mx+a.

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