if $f(\frac{x+y}{2}) =\frac{f(x)+f(y)}{2}$ then find $|f(2)|$

Question

if the following functional equation $$f\bigg(\frac{x+y}{2}\bigg) =\frac{f(x)+f(y)}{2} \quad \text{ holds for all real }~ x ~\text{ and }~ y$$ If$f'(0)$ exists and equals to $-1$ then find $|f(2)|$.

My work

I tried to find $f'(x)$ so that i can intregrate it and then find $f(x)$

$f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$

which gave me this equation:

$f'(x)=\lim_{h \to 0} \frac{f(2x)-2f(x)+1+f(2h)-1}{2h}$

I could not proceed further

and the answer is $2$

Answer 1

since $f'(0)=-1$

we get $f'(x)=-1$

we can integrate both sides and put $x=2$

(and also sorry for mistakes in question. I'm new, so this all was a bit hard and I made silly mistakes)

so we get $|f(x)|=2$

Answer 2

If the brackets are not the greatest integer function, then, It's Jensen's Functional Equation and the solution is $f(x) = ax + b~$ for real $a~$ and $~b.$

Therefore from the given, we have

$$ f(0) = b = 0 ~\text{ and }~ f'(x) = a = - 1$$ Therefore the function is $$ f(x) = -x \implies \lvert f(2)\rvert = 2 $$

Answer 3

$$f'(x) = -1$$ Integrate both sides $$f(x) = -x $$ $$|f(2)| = 2 $$