Question

Y=1cosx+2cos2x+3cos3x.....+99cos99x ,find maximuk value of y.

Answer 1

Use identity cos(2x) = -1+2(cos(x))^2. If cos(x) = t, we have to look at the function f(t) = -(1/2)(2t^2–1) +t = -t^2+t+1/2. Note that -1<=t<=1. f(t) = -t^2+t+1/2 = -(t-1/2)^2+3/4. The maximum of the function is 3/4, it occurs at t=1/2. The minimums of f(t) can occur only at an end of the interval -1<=t<=1. f(-1) = -3/2 f(1) = 1/2. So the minimum is -3/2 . Answer: 3/4 and -3/2