Question

for how many values of x will the tangent lines to y= 4sin x and y= x^2/2 be parallel.

Answer 1

For the tangent lines to be parallel, the values of the derivatives must be equal, so
y = 4sinx gives y' = 4cosx
y = (x^2)/2 gives y' = x
Solving x = 4cosx is not easy. Your best bet is to graph y =x and y = 4cosx and see how many times the graphs intersect. I don't have a graphing calculator handy, but I suspect that there will be three points of intersection, one between 0 and pi/2, one between -pi/2 and -pi, and a third one between -pi and -3pi/2. Don't take my word on it; check it out for yourself.