1. ## Simple Factoring

So I have this equation
f" (x) = -2sinx - 4sin2x
My question is what can I factor out of this equation? my guess is - 2sin or is it just -2?? I'm confused....

2. ## Re: Simple Factoring

Recall that sin(2x) = 2sin(x)cos(x)

3. ## Re: Simple Factoring

So I'm asked to find the points of inflection and to discuss the concavity of the graph of the function
In order to do that I know i need to take the first and second derivative... so here's the work that I have so far

- Original Function
f(x) = 2sinx + sin2x

- First Derivative
f' (x) = 2cosx + 2cos2x

- Second Derivative
f" (x) = -2sinx - 4sin2x

I only needed to take up to the second derivative.... if I take the sin2x from 4sin2x and turn it into 2sin(x)cos(x) wouldn't I be taking the third derivative when i only needed to take the second derivative??

4. ## Re: Simple Factoring

In the case of substituting sin(2x) with 2sin(x)cos(x), all we would be using is a trigonometric identity to simplify.

f''(x) = -2sin(x) - 4sin(2x) = -2sin(x) - 4*(2sin(x)cos(x) = -2sin(x) - 8sin(x)cos(x) = -2sin(x) [ 1 + 4cos(x) ]

5. ## Re: Simple Factoring

So to find the inflection points i need to set x equal to zero... heres what i got

-2sinx = 0
sinx = -2

and the other one is...

1 + 4cos(x) = 0
4cos(x) = 1
cos(x) = -1/4

So when i set both x equal to zero i got sinx = -2 and cos(x) = -1/4
NOW at this point I'm lost.... I'm not familiar with this type of situation... usually i'm used to seeing sinx = 1/2, and i know that equals to Pi/6 and 5Pi/6 in the unit circle and just plug it in a number line and check for concavity... but now i'm confused, what do i do now?

6. ## Re: Simple Factoring

Simple arithmetic error
-2sin(x) = 0
sin(x) = 0 [divide by -2 on both sides]

which makes sense since the zeroes wouldn't change if you were to apply a vertical stretch by a factor of -2.

For cos(x) = -1/4, we know that a solution exists since -1/4 is in [-1,1] so apply cos^-1 to both sides to get your solution.

7. ## Re: Simple Factoring

okay so since sinx = 0, then that corresponds to Pi and 0 in the unit circle
and cosx is equal to -1 and 1? if it is then cos of -1 corresponds to Pi while cos of 1 corresponds to 0... am i right?

8. ## Re: Simple Factoring

okay so since sinx = 0, then that corresponds to Pi and 0 in the unit circle
and cosx is equal to -1 and 1? if it is then cos of -1 corresponds to Pi while cos of 1 corresponds to 0... am i right?

9. ## Re: Simple Factoring

No. [-1,1] is the interval in which cos^-1 is formally defined. For example, if we had cos^-1 = -5, then the solution would not exist. but since -1/4 is in the interval [-1,1], then the solution does exist.