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

(a) Prove that \(f\left( x \right)\) is concave if and only if \(f\left( x* \right)+\left( x-x* \right)f'\left( x* \right)\ge f\left( x \right)\)

(b) Given that \(f\left( x \right)\) is concave, prove that x* is a global maximum of \(f\left( x \right)\) if and only if \(f'\left( x* \right)=0\).

(c) 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.