Critical Numbers and Intervals for f(x)=sin2x over [pi, 2pi]

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In summary, the function f(x)=sin2x has a derivative of 2cos2x. To find the critical numbers, we set the derivative equal to 0 and solve for x. Using the unit circle, we find that the angles in the interval from pi to 2pi where cos(A)=0 are pi/2 and 3pi/2. By substituting 2x for A, we can solve for x and determine that the critical numbers are pi/4 and 3pi/4. The function is increasing on the interval [pi/4, 3pi/4] and decreasing on the interval [3pi/4, 5pi/4].
  • #1
renob
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Homework Statement


Find all the critical numbers for f(x)=sin2x and the open intervals on which the function is increasing or decreasing over the interval [pie, 2pie].


The Attempt at a Solution


I found the derivative to be:
2cos2x

I need help solving the derivative for 0:
0=2cos2x
 
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  • #2
First consult the unit circle to answer the following question: For what angles in the interval from Pi to 2*Pi is cos(A)=0?
 
  • #3
Tom Mattson said:
First consult the unit circle to answer the following question: For what angles in the interval from Pi to 2*Pi is cos(A)=0?

that would be pi/2 and 3pi/2. from there do I multiply them by 2?
 
  • #4
2 cos(2x)= 0 is, of course, the same as cos(2x)= 0.

If you let A= 2x, then you have cos(A)= 0 so , as you say, A= pi/2 or 3pi/2.
Now remember that A= 2x.
How would you solve 2x= pi/2 and 2x= 3pi/2?
 

FAQ: Critical Numbers and Intervals for f(x)=sin2x over [pi, 2pi]

1. What are critical numbers?

Critical numbers are values in a function where the derivative is equal to zero or undefined. They are important in finding the maximum and minimum points of a function.

2. How do I find critical numbers?

To find critical numbers, you must first take the derivative of the function and then set it equal to zero. Solve for the variable to find the critical numbers.

3. Why are critical numbers important?

Critical numbers help us identify the maximum and minimum points of a function, which are important in understanding the behavior and shape of the function.

4. Can there be more than one critical number in a function?

Yes, there can be multiple critical numbers in a function. This is because there can be more than one point where the derivative is equal to zero or undefined.

5. Are critical numbers the same as inflection points?

No, critical numbers and inflection points are not the same. Critical numbers help us find the maximum and minimum points of a function, while inflection points are where the concavity of the function changes.

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