Let \( f \) be derivable function \( \forall x \in R \) such that \( f\left(\frac{x+y}{2}\right)...

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Let \( f \) be derivable function \( \forall x \in R \) such that \( f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2} ; \forall x, y \in R \). If \( f^{\prime}(0)=-1 \) and \( f(0)=1 \), then :
(a) \( 2 f^{-1}(x)=f(x) \)
(b) \( f^{-1}(x)=2 f(x) \)
(c) \( f^{-1}(x)=-f(x) \)
(d) \( f^{-1}(x)=f(x) \)
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