The Chain Rule problem find the points on the horizontal tangent line

Question

Find all points on the graph of the function f(x) = 2 cos(x) + (cos(x))^2 at which the tangent line is horizontal. Consider the domain x = [0,2π).

to find

 (      ,      ) (smaller x value)

  ( pi , -1 ) (larger x value)

 

I found the larger solution, but not the smaller one. I know cosx = -1

I think it should be pi. The webassign says it's wrong,

and also I guessed -pi/2 and pi/2 as x . neither is right. Please help, Thanks!

Answer 1

f'(x)=-2sin(x)-2cos(x)sin(x). f'(x)=0 when the tangent line is horizontal. This means that sin(x)(1+cos(x))=0, so x=0, (pi), 2(pi). The corresponding values of f(x) are given by f(2(pi))=f(0)=2+1=3; f((pi))=-2+1=-1. So the points are (0,3), ((pi),-1), (2(pi),3).