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How to find the analytical solution for this optimization problem? $$ \begin{align} & \underset{x,y,z}{\text{maximize}} & (1+\frac{x}{1+z})(1+\frac{y}{1+z})\\ & \text{such that} & 0\leq x\leq x^+\\ && 0\leq y\leq y^+\\ && 0\leq z\leq z^+\\ && x,y,z\in\mathbb{R}\\ \end{align} $$ where $x^+$,$x^+$ and $z^+$ are positive numbers.

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$$(1+\frac{x}{1+z})(1+\frac{y}{1+z})\le (1 + x) (1 +y) \le (1 + x^+)(1 + y^+)$$

The function achieves the maximum at $(x^+, y^+, 0)$

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