Question

Hi I need help on this question!

Find all points on the graph of the function f(x)=2sinx+sin2x,0≤x<2π at which the tangent line is horizontal. Please list the x-values below separating them with commas.

Answer 1

Hey Soccer :)

The slope of these tangent lines is given by the derivative function.
`horizontal tangent lines` refers to a line with ZERO slope.

So we want to know when our derivative function is equal to zero.

Taking a derivative gives us:\[\Large\rm f'(x)=2\cos x+2\cos2x\]Any trouble with taking derivative?
After that, you want to set your derivative equal to zero, and then solve for x to find out which x values will give you a horizontal tangent line.

Answer 2

\[\Large\rm 0=2\cos x+2\cos2x\]

Answer 3

So it looks like you need to apply your Cosine Double Angle Formula, and then you'll have a quadratic in cosine.

Answer 4

Oh let's also divide that 2 out of the problem:\[\Large\rm 0=\cos x+\cos 2x\]

Answer 5

Applying Double Angle Identity:
\[\Large\rm 0=\cos x+2\cos^2x-1\]

Answer 6

And then solve it like a quadratic :)
Maybe throw it into the Quadratic Formula.

Hopefully that'll get you moving in the right direction.

Answer 7

thank you sooo much :)