Find the points on a horizontal line

[ahazen]

Question: 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).

f'(x)=2cos x+2sinxcosx...i think

No idea where to go after this...

 

[Office_Shredder]

Ok, if the tangent line is horizontal, what does f'(x) have to be?

 

[ahazen]

f'(x)=2cosx +2sinx*(sinx)d/dx
=2cosx +2sinx*cosx
=cos (2+2sinx)

 

[MysticDude]

Hi there, the first thing that I would do is to factor out the 2cos(x).

1. 2cosx(sinx + 1) = 0

Note that this has to be equal to zero because when a tangent line is horizontal it has to be zero.

So all you have to do is find when sinx = -1 OR cosx = 0 on [0,2π)
So x can either be (3π)/2 (that's when sin(x) = -1) or π/2 (that's when cos(x) = 0).

I double checked this with my graphing calculator and these points are also the relative maxima, where the tangent line is 0.

 

[ahazen]

Thank you so much:) I really appreciate it:)

 

[MysticDude]

And just to show it it visually so you can fully understand:[PLAIN]http://img818.imageshack.us/img818/2223/mathp.png

 

[ahazen]

Oh, ok:) that makes sense:) Thank you:)