(1 pt) Find all critical values for the function
f(x) = 6 x^3 - 18 x + 1,
and then list them (separated by commas) in the box below.
List of critical numbers:
critical values are where f' = 0 or undefined.
f'(x) = 18x^2-18 = 18(x^2-1)
So, where is f' = 0?
To find the critical values of a function, we need to determine the x-values where the derivative of the function equals zero or is undefined.
First, let's find the derivative of the function f(x) = 6x^3 - 18x + 1 using the power rule of differentiation. Taking the derivative, we get:
f'(x) = 18x^2 - 18
To find where the derivative equals zero, we set f'(x) = 0 and solve for x:
18x^2 - 18 = 0
We can factor out the common factor of 18:
18(x^2 - 1) = 0
Now, we set each factor equal to zero:
x^2 - 1 = 0
(x - 1)(x + 1) = 0
From this equation, we find two critical values: x = -1 and x = 1.
So, the critical numbers for the function f(x) = 6x^3 - 18x + 1 are -1 and 1.
Therefore, the list of critical numbers for the function is: -1, 1.