Calculus Question- Find the open interval

Question

Find the open interval(s) on which the function f(x) = cos^2(2x)  is increasing in the interval (0,2π). Round
numerical values in your answer to three decimal places.

Answer 1

Hi Kayla.

 

To find the places where a function is increasing, you first need to take its derivative and figure out where on the interval it equals 0. Once you determine the zeros, use a number line to figure out if its positive or negative in that region.

 

First the derivative.

 

f(x) = cos²(2x)

 

f'(x) = 2*cos(2x)*2

 

= 4*cos(2x)

 

Set this equal to 0.

 

0 = 4*cos(2x)

 

cos(2x) = 0

 

We know that cos(θ) = 0 when θ = pi/2, 3pi/2, 5pi/2, 7pi/2... etc

 

We don't have θ, though, we have 2x. So we need to figure out the values of x which make the term equal to those values.

 

2x = pi/2

x = pi/4

 

2x = 3pi/2

x = 3pi/4

 

2x = 5pi/2

x = 5pi/4

 

2x = 7pi/2

x = 7pi/4

 

We're still in our interval of (0,2pi) so let's keep going.

 

2x = 9pi/2

x = 9pi/4

 

Uh oh. Now we're beyond our interval. We'll stop at x = 7pi/4

 

If we were to draw a number line, we'd have 5 different intervals to investigate:

 

1. (0, pi/4)

2. (pi/4, 3pi/4)

3. (3pi/4, 5pi/4)

4. (5pi/4, 7pi/4)

5. (7pi/4, 2pi)

 

Let's pick a test point in each interval.

 

For interval 1, I'll choose pi/8.

 

Plug x = pi/8 into f'(x).

 

f'(pi/8) = 4*cos(2*pi/8) = 4*cos(pi/4)

 

This is in the first quadrant, where cosine is positive.

 

Interval 2: choose pi/2

 

f'(pi/2) = 4*cos(2*pi/2) = 4*cos(pi) = -4, so it's negative.

 

Interval 3: choose pi.

 

f'(pi) = 4*cos(2*pi) = 4*cos(2pi) = 4, so it's positive.

 

Interval 4: choose 3pi/2

 

f'(3pi/2) = 4*cos(2*3pi/2) = 4*cos(3pi) = -4, which is negative.

 

Interval 5: choose 15pi/8

 

f'(15pi/8) = 4*cos(2*15pi/8) = 4*cos(15pi/4), which is in the 4th quadrant, so it's positive.

 

If the derivative is positive, it means your function is increasing. If the derivative is negative, it means the function is decreasing. Since you want to know when it's increasing, your intervals of interest are Intervals 1 and 3 and 5.

 

So, your final answer is:

 

(0,pi/4)U(3pi/4,5pi/4)U(7pi/4,2pi)

 

Hope this makes sense. Let me know if anything I did was confusing.