locate the critical points of the following function. then use the second derivative test to determine…

locate the critical points of the following function. then use the second derivative test to determine whether they correspond to local maxima, local minima, or neither. f(x)=x^3 - 9x^2 what is(are) the critical point(s) of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) is(are) x = . (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. there are no critical points for f.

locate the critical points of the following function. then use the second derivative test to determine whether they correspond to local maxima, local minima, or neither. f(x)=x^3 - 9x^2 what is(are) the critical point(s) of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the critical point(s) is(are) x = . (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. there are no critical points for f.

Answer

Explanation:

Step1: Find the first - derivative

Differentiate $f(x)=x^{3}-9x^{2}$ using the power rule. The power rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. So, $f^\prime(x)=3x^{2}-18x$.

Step2: Set the first - derivative equal to zero

Set $f^\prime(x)=0$, so $3x^{2}-18x = 0$. Factor out $3x$: $3x(x - 6)=0$. Then, by the zero - product property, $3x=0$ or $x - 6=0$. Solving these equations gives $x = 0$ and $x=6$.

Step3: Find the second - derivative

Differentiate $f^\prime(x)=3x^{2}-18x$ to get $f^{\prime\prime}(x)=6x-18$.

Step4: Evaluate the second - derivative at the critical points

For $x = 0$, $f^{\prime\prime}(0)=6\times0-18=-18<0$. So, $f(x)$ has a local maximum at $x = 0$. For $x = 6$, $f^{\prime\prime}(6)=6\times6-18=36 - 18 = 18>0$. So, $f(x)$ has a local minimum at $x = 6$.

Answer:

A. The critical point(s) is(are) $x = 0,6$