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# Workouts

## Shape of Functions

• 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.

## Misc

• 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.