Question: Let \(m\left( \alpha ,y \right)\) be defined as the minimum value of \(\alpha x\) subject to \(g\left( x \right)\ge y\), where \(\alpha ,x\in R_{++}^{n}\), \(y\in {{R}_{++}}\) and g(x) is strictly monotonic increasing and quasi-concave. Prove that \(m\left( \alpha ,y \right)\) is (i) non-decreasing in \(\alpha \) and y and (ii) concave in \(\alpha \). Then, given that g(x) is homogeneous of degree k, derive the corresponding form of \(m\left( \alpha ,y \right)\)