Hello, please help me. I needed to find the derivative of f(x)= x^4 -2x^3 + x +1. I got f'(x)= 4x^3 -6x^2 +1. I would like to figure out the critical numbers so I can figure out where it increases and decreases, but I have no idea how to factor it... Is it even possible?
Thank you for your time.
tried x = ± 1 , no good
tried ± 1/2 and ahhh! x = 1/2 is a root
so (2x-1) is a factor
long division:
4x^3 -6x^2 +1 = (2x-1)((2x^2 - 2x-1)
2x^2 - 2x-1 = 0 gave me
x = (1 ± √3)/2
so there are 3 critical points to worry about.
take it from here.
this factors into
2(2x-1)(x^2 - 2x - 1)
x = 1/2 or (1±√3)/2
Having that, you can find where f'<0 or f'>0
Oooooh! Thank you two so much :) I'm more confident in regards to my test now, thank you
Hello! To find the critical numbers of a function, you need to find the values of x where the derivative is either zero or undefined. In your case, you have already found the derivative of the function f(x) as f'(x) = 4x^3 - 6x^2 + 1.
To determine the critical numbers, you need to find the values of x that make f'(x) = 0. In this case, you have a polynomial equation to solve for x: 4x^3 - 6x^2 + 1 = 0.
Factoring a polynomial equation with a degree higher than 2 can be a bit challenging. However, in this case, the equation doesn't appear to factor easily. So, we need to resort to an alternative method to find the critical numbers.
One approach is to use numerical methods like graphing or a calculator to find the x-values at which the derivative is zero or near zero. You can graph the derivative function f'(x) and look for the x-intercepts or use a calculator or online tool that can find the root(s) of a function.
Once you find the approximate values of the critical points, you can use them to further analyze the behavior of the original function f(x) and determine where it increases or decreases.
I hope this explanation helps you understand how to find critical numbers when factoring is difficult or not feasible. Let me know if you need further assistance!