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 find f(x). f(x)=6x - 18 what is/are the local maximum/maxima of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/maxima of f is/are at x = 0. (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. there is no local maximum of f. what is/are the local minimum/minima of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local minimum/minima of f is/are at x =. (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. there is no local minimum of 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 find f(x). f(x)=6x - 18 what is/are the local maximum/maxima of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/maxima of f is/are at x = 0. (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. there is no local maximum of f. what is/are the local minimum/minima of f? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local minimum/minima of f is/are at x =. (use a comma to separate answers as needed. type an integer or a simplified fraction.) b. there is no local minimum of f.

Answer

Explanation:

Step1: Find the first - derivative

Differentiate $f(x)=x^{3}-9x^{2}$ using the power rule. $f^\prime(x)=3x^{2}-18x = 3x(x - 6)$. Set $f^\prime(x)=0$, then $3x(x - 6)=0$, so the critical points are $x = 0$ and $x=6$.

Step2: Apply the second - derivative test

We know $f^{\prime\prime}(x)=6x - 18$. When $x = 0$, $f^{\prime\prime}(0)=6\times0-18=-18<0$. By the second - derivative test, $f(x)$ has a local maximum at $x = 0$. When $x = 6$, $f^{\prime\prime}(6)=6\times6-18=18>0$. By the second - derivative test, $f(x)$ has a local minimum at $x = 6$.

Answer:

A. The local maximum/maxima of f is/are at x = 0. A. The local minimum/minima of f is/are at x = 6.