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I have the following equation $$f(a)=\text{argmin}_{t\in\mathbb{R}} \frac{1}{2}(1-a)(b-t)^2+a| b-t |, a\in[0,1]$$

How does one prove that the function is monotone in a and find the limits for it(effectively finding its image).

I was given the hint that one can use differentiation. I have however never worked with differentiation in the context of argmin functions. Heuristically speaking if one ignore the argmin one gets positive derivative for $| b-t |\in[-2,2]$. For minimization it makes sense for b and t to be "close". I don't really know how to formalize it.

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If the problem statement is written out correctly, it appears that $t = b$ is the unique minimizer of $\frac{1}{2} (1-a) (b-t)^2 + a \left\lvert b - t \right\rvert$ when $a \in [0,1]$ (since the above function is nonnegative in this case). Therefore, $f(a) = b$ is well-defined and trivially monotonic on $[0,1]$.

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