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

Answer 1

f'(x) = 2cosx + 2sinx(cosx)
= 0 when the tangent is horizontal

2cosx(1 + sinx) = 0
cosx = 0 or sinx = -1

if cosx = 0
x = π/2 or 3π/2
f(π/2) = 2(1) + 1 = 3
f(3π/2) = 2(-1) + 1 = -1
so we have two points, (π/2 , 3) and (3π/2 , -1)

if sinx = -1
x = 3π/2 giving us the same point as above

there are two points (π/2 , 3) and (3π/2 , -1)

Answer 2

Thanks!!! I was getting the first point wrong!