Let the value function be defined as in (6.9) with respect to a utility function (u: mathbb{R}_{+} ightarrow mathbb{R}) strictly increasing and concave. Prove

Let the value function be defined as in (6.9) with respect to a utility function \(u: \mathbb{R}_{+} \rightarrow \mathbb{R}\) strictly increasing and concave. Prove that the value function is strictly increasing and concave in its first argument (wealth).