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I am working on a linear static optimization problem. I found a solution to the problem. However, I want to formally check the solution existence. I tried some methods but I don't know if it is enough to check those. I will note below the problem and the methods I tried. I will appreciate your comments.

$\underset {a_i or G_i^L} {max}$ $\alpha$ ln $C_i$ + $\beta$ ln $G_i^L$ + $\beta$ ln $G_i^C$



$G_i^C$ = $\sum \limits_{i=1}^n$[(1-fd)t$Y_i$]$ p_i \over \sum \limits_{i=1}^n p_i$

where 0<$\alpha$<1 and 0

0$<fd<$1, 0$<t<$1, 0$<p_i<$1, $a_i,Y_i >$0

what I tried is called Inada conditions: checking the limit of the first derivatives w.r.t. $C_i$, $G_i^C$, and $G_i^L$. When that limit goes to infinity the derivatives goes to zero. This satisfies the Inada conditions. I also checked that this objective function is concave and increasing in $C_i$, $G_i^C$, and $G_i^L$. Second derivatives are positive.

Do I have to check the Weierstrass Theorem? I have a solution to this problem but the substitutable equalities are not compact.

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