Question: Prove that \(f:{{R}^{n}}\to {{R}^{n}}\) is concave if and only if \(f\left( \sum\limits_{i=1}^{m}{{{\lambda }_{i}}{{x}^{i}}} \right)\ge \sum\limits_{i=1}^{m}{{{\lambda }_{i}}f\left( {{x}^{i}} \right)}\) for all \({{x}^{i}},...,{{x}^{m}}\in {{R}^{n}}\) and for all \({{\lambda }_{1}},...,{{\lambda }_{m}}\), where \(\sum\limits_{i=1}^{m}{{{\lambda }_{i}}}=1\), and \({{\lambda }_{i}}\ge 0\).

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