Homework 7 trig...problem 4


Find the second smallest positive xx-value where the graph of the function f(x)=x+5sin(2x) has a horizontal tangent line.

im not sure how to give an exact x value. when graphed the second smallest x value with a slope of zero is just past 2pi//3 the decimal value (2.306,-2.699).

if any one can help out that would be awesome!


My problem was f(x)= x+3sin(3x).

I am getting(1/3)cos^(-1)(-1/9) but its wrong so sorry.


First off, my practice problem did not have the 2 in it, which basically means that I didn’t think you’d need to use the chain rule to do it. Furthermore, you need to know about inverse trig functions for this, too. I really wasn’t worried about that even though we haven’t reviewed it yet, since it is pre-requisite material. So, I might consider giving everyone credit for this problem.

Having said all that, let’s throw caution to the wind and give it a go!

The tangent line is horizontal when f'(x)=0 and

f'(x) = 1+2\times5\cos(2x) = 1+10\cos(2x)

If we set that equal to zero and try to solve for x, we get


We might be tempted to apply the \arccos to both sides to get

2x = \arccos\left(-\frac{1}{10}\right) \: \text{ or } \: x = \frac{1}{2}\arccos\left(-\frac{1}{10}\right).

Let’s take a look at a graph, though:


Now, the red dot near \pi/4 is actually x=\arccos(-1/10)/2, that we found using the arccos. You actually want the green dot, though. It’s pretty clear from the symmetry that the blue dot is at -\arccos(-1/10)/2. Furthermore, the green dot is one period away from the blue dot and the period of \cos(2x) is \pi. So, I guess the answer is:

x = \pi - \arccos(-\pi/10)!