(See Solution) Given that f: R_++^n \rightarrow R, prove that f(x)=(∑_i=1^n α_i x_i+β)/((∑_i=1)^n γ_i x_i+δ)^2, where β,

Question: Given that $f: R_{++}^{n} \rightarrow R$, prove that

$f(\mathrm{x})=\frac{\sum_{i=1}^{n} \alpha_{i} x_{i}+\beta}{\left(\sum_{i=1}^{n} \gamma_{i} x_{i}+\delta\right)^{2}}$, where $\beta, \delta>0, \alpha, \gamma \in R_{++}^{n}$, is quasi-concave;
$f(\mathrm{x})=\frac{h(\mathrm{x})}{g(\mathrm{x})}$, where $h(\mathrm{x})>0$ is concave and $g(\mathrm{x})>0$ is convex, is quasi-concave; and
$f(\mathrm{x})=\ln \left(\sum_{i=1}^{n} x_{i}\right)$ is both quasi-concave and quasi-convex.

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(See Solution) Given that f: R_++^n \rightarrow R, prove that f(x)=(∑_i=1^n α_i x_i+β)/((∑_i=1)^n γ_i x_i+δ)^2, where β,

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