Let $Y \in R^n$ and $\mathrm X \in M_{np}(\bf R)$ such that $$\sum_{i=1}^ny_i=0 \text{ and } \sum_{i=1}^nx_{ij}^2 = 1, \text{ for } j = 1,\dots,p.$$ For $\beta \in \bf R^p$, define $\mathcal L(\beta) = ||Y-\mathrm X\beta||^2 + \lambda||\beta||^2 + \mu|\beta|$, for $\lambda, \mu > 0.$
Prove that the minimum of $\beta_j \mapsto \mathcal L(\beta_1, \dots, \beta_j, \dots, \dots, \beta_p)$ is reached at $$\beta_j = \frac{R_j}{1+\lambda}\left(1-\frac{\mu}{2|R_j|}\right) \text{ with } R_j = \mathrm X_j^T\left(Y - \sum_{k\ne j}\beta_k \mathrm X_k\right).$$
I solved for the partial derivative of $\mathcal L$ with respect to $\beta_j \ne 0$: $$\partial_j\mathcal L(\beta) = 2\left((1+\lambda)\beta_j - R_j + \frac{\mu}{2}\text {sign}(\beta_j)\right) $$ But it did not help me to go forward...
Can anyone help me for the following?