Just so you know, this is a homework question, and I basically need help with the steps to solve this problem. I understand what it's asking; however, my attempts haven't worked out, and I'm probably over-complicating it. Anyway, here it is:
I have a space $C([-\pi/2,\pi/2])$ with the supremum norm: $||f||_\infty = \sup\{|f(x)| : x \in [-\pi/2,\pi/2]\}$ and I need to find the closest linear function $g(x) = cx$ closest to the function $f(x) = \sin(x)$ w.r.t. this norm.
In other words, it appears that I need to find $c$ that minimizes the distance between these two curves, or that generates the smallest least upper bound for the function $h(c) = |\sin(x)-cx|$ for $x \in [-\pi/2,\pi/2]$.
I thought this would be the same as minimizing the area between these two curves as a function of $c$, but it's not optimal to find the points of intersection of these functions as it would be dependent on $c$.
Any ideas?