Question

Find the x-coordinates of all points on the curve
f(x) = sin 2x − 2 sin x
at which the tangent line is horizontal.

Answer 1

1º differentiate f(x):
f'(x)=2cos2x+2cosx
2º a line is horizontal when it's slope is equal to 0. Actual value of the slope of tangent line to a curve f(x) is it's derivative at the point in question. So make the derivative of f(x) equal to 0 and solve resulting equation:

2cos2x+2cosx=0
2(cos^2x-sin^2x)+2cosx=0
2(cos^2x-1+cos^2x)+2cosx=0
4cos^2x+2cosx-2=0
2cos^2x+cosx-1=0
now let cosx=t, then
2t^2+t-1=0
solve this for t, and later substitute t=cosx and find value of x.

Answer 2

Ok, I solved for t and I got cosx= 1/2 and cosx=-1 so is this the answer?

Answer 3

no. Now solve for x

Answer 4

I got x= \[\pi/3\] and \[\pi \]

Answer 5

there is one more point:
5Pi/3

Answer 6

so all points of type x=Pi +n2Pi, x=Pi/3+n2Pi, x=5Pi/3+n2Pi, will have derivative equal to 0

Answer 7

The answer is still wrong?

Answer 8

?