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. (Enter your answers as a comma-separated list. Use n to represent any integer.)

Answer 1

is it \(f(x)=\sin(2x)-2\sin(x)\) ?

Answer 2

I suppose so. The question just has sin 2x - 2 sin x, but that is how I'd read it yeah

Answer 3

take the derivative, set it equal to zero and solve

Answer 4

\[f'(x)=2\cos(2x)-2\cos(x)\]

Answer 5

\[2\cos(2x)-2\cos(x)=0\]
\[\cos(2x)=\cos(x)\]

Answer 6

hmm how to solve
any even multiple of \(\pi\) works, but i am sure there are more
maybe start with
\[\cos(2x)-\cos(x)=0\] then rewrite as
\[2\cos^2(x)-\cos(x)-1=0\] and solve the quadratic for cosine

Answer 7

Pretty lost. How do I know if the tangent is horizontal or even where it is?

Answer 8

tangent horizontal = derivative is zero

Answer 9

solve
\[2\cos^2(x)-\cos(x)-1=0\]
\[(2\cos(x)+1)(\cos(x)-1)=0\]\[\cos(x)=-\frac{1}{2}\] or \[\cos(x)=1\]

Answer 10

I don't get it. How many values should the answer be? is the answer where those equal 0?