4
$\begingroup$

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?

1 Answers 1

2

To begin with, fix $c \in\mathbb{R}$ and find the maxima of the function $g(x)=|cx-sinx|.$ One way to do this is to differentiate $g(x)$ with respect to $x.$ Notice that you can do the differentiation at all points except where $cx=sinx.$ Call this maxima $m(c),$ explicitly highlighting the dependence on $c.$ Now try finding minimum of $m(c).$

  • 1
    Be careful. $arccos$ is defined only when $-1 \leq c \leq 1$. This suggests you may need to split your problem into different cases based on $c.$2012-02-07