Seung hyun H.

asked • 07/30/23

How many critical numbers does f have?

A formula for the derivative of a function f is given. How many critical numbers does f have?


f'(x)= 100cos^2(x) / 10+x^2 -1


I know that the critical point exists where f'(x) does not exist and the x values that make f'(x)=0.

But I don't know how to apply it this problem because of 100cos^2(x).


I want to know the process and the answer.

Doug C.

tutor
You might want to clarify your denominator. And when you use a forward slash to separate numerator and denominator of a fraction, use a grouping symbol if there is more than one term: 100cos^2(x) / (x^2 + ???)
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07/30/23

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Doug C. answered • 07/30/23

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The 1st derivative equals zero when the numerator is equal to 0.


100 cos2(x) = 0

cos2(x) = 0


cos(x) = ±0


x = cos-1(0)


There are infinitely many numbers that have a cosine of 0 where one of them is π/2. Any angle with terminal side along the y-axis has a cosine of 0.


So x can be any of π/2 + kπ, where k is any integer.


Take a look here (where the denominator of the given function was taken as (x^2+10x-1).


desmos.com/calculator/u8pgt50ui4


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Seung hyun H.

Sorry, I mean the equation of f'(x) is equal to (100cos^2(x) / 10+x^2) -1. How can I find the critical point of this? Now I know the 100cos^2x is equal to 0 when x is π/2.
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07/30/23

Doug C.

tutor
Did you visit the Desmos graph I provided? There are infinitely many critical points (points where the first derivative is equal to zero). It is still not clear what the representation for f'(x) is. It looks like you mean: f'(x) = [(100cos^2(x)) / (10+x^2)] - 1. If that is the case then that changes the critical numbers. Setting equal to zero and solving for x results in 100cos^2(x)-x^2-10x = 0. The roots there cannot be found algebraically. Have you learned Newton's method? Check this updated graph to see that there are 3 critical numbers (if your derivative definition has been interpreted correctly. desmos.com/calculator/noqdofauby
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07/30/23

Doug C.

tutor
Ahh, I see that the denominator of the function should be (x^2+10), not (x^2+10x). desmos.com/calculator/tf1m44bic1
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07/31/23

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