4. - / 1 points find the critical numbers of the function. (enter your answers as a comma - separated list…

Explanation:

Step1: Find the derivative

Using the power - rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=12x^{3}+36x^{2}-48x$.

Step2: Set the derivative equal to zero

$12x^{3}+36x^{2}-48x = 0$. Factor out $12x$: $12x(x^{2}+3x - 4)=0$.

Step3: Factor the quadratic

Factor $x^{2}+3x - 4=(x + 4)(x - 1)$. So the equation becomes $12x(x + 4)(x - 1)=0$.

Step4: Solve for x

Set each factor equal to zero:

• If $12x=0$, then $x = 0$.
• If $x + 4=0$, then $x=-4$.
• If $x - 1=0$, then $x = 1$.

Answer:

$-4,0,1$