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Function with midpoint averaging property

High school · ●●●○○ 50 pts

Let ff be a differentiable function satisfying f(x+y2)=f(x)+f(y)2f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} for all real xx and yy. Given that f(0)=1f(0)=1 and f′(0)=−1f'(0)=-1, find f(2)f(2).

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functional equationsderivatives