I'm trying to solve the following optimization problem: \begin{equation}\end{equation} \begin{align} \text{argmax}_{\left\{x_i\right\},\left\{y_j\right\}} & \frac{1}{2}\left( \sum_{i=1}^{N} \log_2\left(1+\alpha_i x_i\right) + \sum_{j=1}^{M} \log_2\left(1+ \beta_j y_j\right) \right) \\[4pt] \nonumber \text{subject to: }& \sum_{i=1}^N x_i + \sum_{j=1}^M y_j \leq A \\ & \sum_{j=1}^{M} \log_2\left(1+ \beta_j y_j\right) \leq B \end{align}
To this end, I tried to use the Lagrangian multipliers method. I defined the Lagrangian function \begin{align} \mathcal{L}(x_1,\ldots,x_N,y_1,\ldots,y_M,\lambda_1,\lambda_2) &= \left( \sum_{i=1}^{N} \log_2\left(1+\alpha_i x_i\right) + \sum_{j=1}^{M} \log_2\left(1+ \beta_j y_j\right) \right) \\ & - \lambda_1 \left(\sum_{i=1}^N x_i + \sum_{j=1}^M y_j - A\right) \\ & -\lambda_2 \left(\sum_{j=1}^{M} \log_2\left(1+ \beta_j y_j\right) -B\right) \end{align} and I computed the partial derivatives of the Lagrangian function with respect to generic $x_i$ and $y_j$. Setting them to zero led to \begin{align} & \frac{\partial \mathcal{L}}{\partial x_i} = \frac{\alpha_i}{\alpha_i x_i \log(2) + \log(2)} - \lambda_1 = 0, \\ & \frac{\partial \mathcal{L}}{\partial y_j} = \frac{\beta_j}{\beta_j y_j \log(2) + \log(2)} - \lambda_1 -\lambda_2 \frac{\beta_j}{\beta_j y_j \log(2) + \log(2)} = 0, \end{align} which resulted in \begin{align} & x_i = \frac{1}{\log(2) \lambda_1} - \frac{1}{\alpha_i}, \\ & y_j = \frac{1-\lambda_2}{\log(2) \lambda_1} - \frac{1}{\beta_j}. \end{align} I don't know how to proceed from here onwards. Any help?