When is \(\left(\frac{1}{x}+\frac{1}{y}\right)^\alpha\) quasiconcave on \((x,y)\in\mathbb{R}^2_{>0}\)?

Prove \(\left(1+\frac{1}{x}\right)^{x}\) is monotone on \(x\in\mathbb{R}_{>0}\).

Prove \(\prod_i^n\left(x_i^{\alpha_i}\right)\) is quasiconcave.

- Show there is no smooth profit function where profit and average profit are both maximized at the same interior point \((q>0)\) and where profit is strictly positive \((\pi > 0)\). Construct a discontinuous profit function where there is such an interior optimum.