Thread: How do I find the critical numbers?

1. How do I find the critical numbers?

The polynomial function is: f(x) = x(x-4)^3

I know that you have to find the first derivative of the function, which is:
f'(x): 4x^3 - 36x^2 + 96x - 64
f'(x): x^3 - 9x^2 + 24x - 16

We are supposed to set f'(x)=0, so:

x^3 - 9x^2 + 24x - 16 = 0

I would think that you bring the "-16" to the other side:

x^3 - 9x^2 + 24x = 16

Factor out the x:

x(x^2 - 9x + 24) = 16

But I know that 16 is not one of the critical numbers, so I'm not sure of where I went wrong. If anyone could correct me or provide suggestions, that would be very helpful! Thank you in advance!

2. Re: How do I find the critical numbers?

You have to solve for x, i.e., you must find the numbers x that satisfy f'(x) = 0

3. Re: How do I find the critical numbers?

Originally Posted by misiaizeska
The polynomial function is: f(x) = x(x-4)^3

I know that you have to find the first derivative of the function, which is:
f'(x): 4x^3 - 36x^2 + 96x - 64
f'(x): x^3 - 9x^2 + 24x - 16 No! You can not just throw away common factors. The only reason you would normally be able to do this is if you have set the equation equal to 0, which you have NOT done yet!

We are supposed to set f'(x)=0, so:

x^3 - 9x^2 + 24x - 16 = 0 This is correct NOW.

I would think that you bring the "-16" to the other side: No! Polynomials can only be solved by factorising and using the Null Factor Law!

x^3 - 9x^2 + 24x = 16

Factor out the x:

x(x^2 - 9x + 24) = 16

But I know that 16 is not one of the critical numbers, so I'm not sure of where I went wrong. If anyone could correct me or provide suggestions, that would be very helpful! Thank you in advance!
Try to find a linear factor of your expression that is equal to 0 by substituting in numbers and applying the remainder and factor theorems.

A better alternative would be to apply the product and chain rules to evaluate the derivative, as the common factors would most likely appear.

4. Re: How do I find the critical numbers?

Prove It has given a very good hint, differentiate the function using product rule:
f'(x) = (x-4)^3 + 3x(x-4)^2 now when you equate it equal to 0 you get factors easily