Find the open interval(s) on which the function f(x) = cos^2(6x) is increasing in the interval (0,2.094)

Question

Round to the nearest three decimal places. I'm not sure if i should take the derivative and then solve for 0. Using critical values. I'm very confused and my notes are no help. 

Answer 1

Yes, take the derivative and set it equal to zero.

 

d/dx [cos^2(6x)] = -12cos(6x)sin(6x)

 

Set it equal to zero

 

cos(6x)sin(6x) = 0

 

cos(x) and sin(x) both have period of 2pi

cos(6x) and sin(6x) both have period of pi/3 which is 2pi/6

 

Over that period, sin(6x) crosses zero twice so the period of sin(6x) = 0 is pi/6

and period of cos(6x) = 0 is pi/6.

 

cos(6x) = 0 when x = pi/12, p/12 + pi/6, etc. to 7pi/12 

sin(6x) = 0 when x = pi/6, pi/6 + pi/6, etc. to 2pi/3 (2.094)

 

the intervals are:

 

(pi/12, pi/6), (3pi/12, pi/3), (5pi/12, pi/2), (7pi/12, 2pi/3)

 

You can confirm that the slope is increasing by picking a value in each interval

and checking that the slope is positive.

 

The interval (pi/12, pi/6) is in the 2nd quadrant where cos is - and sin is +

so the slope is + (remember the negative sign in the 1st derivative).

 

The interval (3pi/12, pi/3) is in the 4th quadrant where cos is + and sin is -
so the slope is + 

 

Answer 2

A function is increasing when it has a non-negative slope

 

Take a simple example with 0< x < 2π = 6.28...

For cos(x) a period is 2π, and the function is increasing on the intervals (π, 2π)

For cos²(x) has a period of π and the function is increasing on the intervals (π/2, π) and (3π/2, 2π)

 

In this example the interval is 0 < x < 2.094 ≈ 2π/3

For the function f(x) = cos(6x), the period τ is 6τ = 2π, τ= π/3 is increasing (π/6 , π/3) and (π/2, 2π/3)

 

Finally to answer the question, the function f(x) = cos²(6x) has a period of π/6 and is increasing on the intervals (π/12, π/6), (π/4, π/3), (5π/12, π/2) and (7π/12, 2π.3)

 

These can verified using a graphing calculator, or just make graphs of these functions.  I did more than the problem asked for to illustrate how to find these increasing intervals on similar functions.