Finding critical numbers

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In summary, the critical numbers for g(x) = sqrt(x^2 - 4) are x = 2, x = -2, and x = 0. For f(x) = (1)/(x^2 - 9), the only critical number is x = 0.
  • #1
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Homework Statement


Find all critical numbers of:
g(x)= sqrt(x2-4)
and
f(x)= (1)/(x2-9)


Homework Equations


n/a


The Attempt at a Solution


1) sqrt(x2-4)
simplified to x(x-2)(x+2)-1/2 (please check)
and got zeroes as x=0, x=2, x=-2
and I got confused because if you do g(2) and g(-2) it equals zero. For some reason I can't tell if they are defined or undefined. x=0 works, so that is a critical number. The other two are throwing me off.

2) (1)/(x2-9)
zeroes were x=-3, x=3, x=0. Plugging 3 and -3 back into f(x) gave me undefined, so I'm pretty sure 0 is the only critical number.

If you could please check the second and help with the first that would be great. Thank you
 
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  • #2
About

"1) sqrt(x2-4)
simplified to x(x-2)(x+2)-1/2"

- is that supposed to be the derivative of [tex] \sqrt{\,x^2 -4}[/tex]? if so, it isn't correct.

what is your definition of a critical value? (writing it out can help you see the appropriate path)
 
  • #3
I'm honestly not sure what I did there... I re-did my derivative and found

x/((x2-4)1/2)

which would go to x/(((x+2)(x-2))1/2)

and give zeroes of -2,2,0. -2 and 2 would make the denominator 0 and be undefined, leaving 0 as the only CP. Sound right? Sorry if any errors I did that really quick.
 
  • #4
yes, the derivative is

[tex]
\frac{x}{\sqrt{\, x^2 - 4}}
[/tex]

and this is zero or undefined for [itex] 0 \text{ and } \pm 2 [/itex].

Again, what is your definition of a critical number? (same as critical value)
 
  • #5
Definition is:
x=c is a critical number for f(x) if f(c) is defined and f'(c)=0 or f'(c) is undefined.

So that means 2 and -2 WOULD be CPs?
 
  • #6
tjohn101 said:
Definition is:
x=c is a critical number for f(x) if f(c) is defined and f'(c)=0 or f'(c) is undefined.

So that means 2 and -2 WOULD be CPs?

yes.
 
  • #7
And 0 would not be one because it is undefined in f(x) (sqrt of a negative makes it undefined), correct?

And for the second problem, zero was the only one defined in f(x). Therefore it IS indeed a CP.
 

1. What are critical numbers?

Critical numbers are the values of a function where its derivative is equal to zero or does not exist. These points are important for determining the maximum and minimum values of a function.

2. How do you find critical numbers?

To find critical numbers, you need to take the derivative of the given function and set it equal to zero. Solve for the variable to find the critical numbers. Additionally, you should also check for values where the derivative is undefined.

3. Why are critical numbers important?

Critical numbers are important because they help us identify the maximum and minimum values of a function. This information is useful in optimization problems and understanding the behavior of a function.

4. Can a function have more than one critical number?

Yes, a function can have multiple critical numbers. This occurs when the derivative of the function is equal to zero at more than one point or when the derivative is undefined at multiple points.

5. What is the difference between critical numbers and inflection points?

Critical numbers and inflection points are different concepts. Critical numbers are points where the derivative of a function is equal to zero or undefined, while inflection points are points where the concavity of a function changes. Critical numbers are used to find maximum and minimum values, while inflection points are used to identify changes in the curvature of a function.

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