Question

find all critical values for the function (f(x)=9x^{3}-27x + 5), and then list them (separated by commas) in the box below. list of critical numbers:

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=9x^{3}-27x + 5$ using the power - rule $(x^n)'=nx^{n - 1}$ is $f'(x)=27x^{2}-27$.

Step2: Set the derivative equal to zero

Set $f'(x)=0$, so $27x^{2}-27 = 0$. Divide through by 27: $x^{2}-1=0$.

Step3: Solve the resulting equation

Factor $x^{2}-1=(x - 1)(x + 1)=0$. Then, by the zero - product property, $x-1 = 0$ or $x + 1=0$. So $x=1$ or $x=-1$.

Answer:

$1,-1$