find the critical points of the following function. f(x)=x³/3 - 144x select the correct choice below and, if…

find the critical points of the following function. f(x)=x³/3 - 144x select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) occur(s) at x= (use a comma to separate answers as needed.) b. there are no critical points.

find the critical points of the following function. f(x)=x³/3 - 144x select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) occur(s) at x= (use a comma to separate answers as needed.) b. there are no critical points.

Answer

Explanation:

Step1: Find the derivative

The derivative of $f(x)=\frac{x^{3}}{3}-144x$ using the power - rule $\frac{d}{dx}(x^{n}) = nx^{n - 1}$ is $f'(x)=\frac{d}{dx}(\frac{x^{3}}{3})-\frac{d}{dx}(144x)$. So $f'(x)=x^{2}-144$.

Step2: Set the derivative equal to zero

Set $f'(x) = 0$, then $x^{2}-144 = 0$. This is a difference - of - squares equation, which can be factored as $(x + 12)(x - 12)=0$.

Step3: Solve for x

Using the zero - product property, if $(x + 12)(x - 12)=0$, then $x+12 = 0$ or $x - 12=0$. Solving these equations gives $x=-12$ or $x = 12$.

Answer:

A. The critical point(s) occur(s) at $x=-12,12$