Question: Suppose that \(f\left( x \right)\) has a continuous first derivative for all \(x\in \mathbb{R}\).

1. Prove that \(f\left( x \right)\) is concave if and only if \(f\left( x* \right)+\left( x-x* \right)f'\left( x* \right)\ge f\left( x \right)\)
2. Given that \(f\left( x \right)\) is concave, prove that x * is a global maximum of \(f\left( x \right)\) if and only if \(f'\left( x* \right)=0\).
3. Given that \(f\left( x \right)\) is concave, prove that its set of global maxima is either empty, a singleton or an infinite convex set.

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