Let $f(x) = \begin{cases} A_1 x + B_1 &\mbox{if } x \in [a, x_0] \\ A_2 x + B_2 & \mbox{if } x \in [x_0, b] \end{cases}$ and $f \in C[a, b]$ i.e. f has the "angle" form.
Denote $E_t(f) = ||f(x) - t||_{L_p[a,b]} = (\int_a^b |f(x) - t|^p dx)^{1/p}$.
What is $\inf_{t \in \mathbb{R}} E_t(f)$?
In other words, which of the constants $t$ best approximates the function in the $L_p$ metric?
I can't find the exact value, so it would be great to know if there are any good estimates for this value.