Question: (a) Suppose that \(A(n)\) is a statement for all natural numbers \(n\) and that (i) \(A(1)\) is true and (ii) if the induction hypothesis \(A(k)\) is true, then \(A(k+1)\) is true for each natural number \(k\). Prove that \(A(n)\) is true for all natural numbers \(n\).

(b) Prove that \(f: R^{n} \rightarrow R\) is concave if and only if \(f\left(\sum_{i=1}^{m} \lambda_{i} \mathbf{x}^{i}\right) \geq \sum_{i=1}^{m} \lambda_{i} f\left(\mathbf{x}^{i}\right)\) for all

\(\mathrm{x}^{1}, \ldots, \mathrm{x}^{m} \in R^{n}\) and for all \(\lambda_{1}, \ldots, \lambda_{m}\), where \(\sum_{i=1}^{m} \lambda_{i}=1\) and \(\lambda_{i} \geq 0, i=1, \ldots, m\)

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