4. Let f be a continuous function on [-7, 7] and let m = N be non-zero. Think of as a complicated wave. Our goal is to approximate it by simpler waves of sine. Let c E R. We want to approximate f(x) by csin(mx). To choose c optimally, we must quantify the error in this approximation. The quantity |f(x) - csin(mx)| represents the error between the functions at a single point x = [-,]. Consider 1 " (f(x) csin(x)) dx 2T - Here, (f(x) csin(mx)) 2 is the square error. The integral represents the average value of the square error on [-T,]. (a) For c ER, define Fm(c) to be the above integral. Use Fm to find the optimally chosen constant c = cm R. Give an integral formula for Cm. Hint: Expand the square and write Fm(c) as a quadratic in c. In this question, we treat m as a fixed positive integer. Also what is - sin (mx) dx?