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

1. Prove that \(f\left( x \right)\) is concave if an only if \(f\left( x* \right)+\left( x-x* \right)f'\left( x \right)\ge f\left( x \right)\), for all \(x\) and \(x*\in R\).
2. Given that \(f\left( x \right)\) is concave, prove that \(x*\) is a global maximum of \(f\left( x \right)\) if an only if \(f'\left( x* \right)=0\).
3. Given that \(f\left( x \right)\) is strictly concave, prove that it cannot posses more than one global maximum.

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