I have the function $f:\left [ 0,\pi \right ]\rightarrow \mathbb{R}, x \mapsto \sin(x)$. How can I choose two points $x_0, x_1 \in \left [ 0,\pi \right ]$ for my polynomial interpolation such that I get a polynomial $p$ of degree $1$ that minimizes the pointwise-error, i.e. $\min\left \| f-p \right \|_{\infty}$.
My thoughts:
Just by taking a closer look at the plot of $\sin(x)$, I think it becomes quite obvious that $x_0$ and $x_1$ have to be chosen such that $p(x) = \frac{1}{2}$.
However, I have no idea how to proof this. I'd really appreciate if someone could show me how to proof this or give me some advice/hints.
Thanks in advance!