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I am trying to get the analytical closed form solution for $d^*$ which maximizes an expression $e^{-\lambda d}(z-v-cd)^{\beta} + (1-e^{-\lambda d})(z-cd)^{\beta}$, where $\lambda, z, v, c, \beta$ are constants. Because of the presence of $\beta$, I doubt that analytical closed form solutions exist. However, is it possible to at least get approximate solutions? When I searched for similar questions, I came across some suggestions to use Lambert W function. Is that applicable in this particular context? Any suggestion is appreciated. Thank you.

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