I have the following function: $$f(a)=\text{argmin}_{t\in\mathbb{R}} \left\{\sum_{i=1}^n\frac{1}{2}(1-a)(b_i-t)^2+a| b_i-t |\right\}\quad ,a\in[0,1]$$
Is there a way to show that the function is monotone in $a$?
Edit: for $a=0$ we have the mean of $\beta_i$s and for $a=1$ we have the median. I'm interested in the cases when $a\in(0,1)$, in particular that the function is bounded by the mean and median.