sketch the graph of the following function. list the coordinates of where extrema or points of inflection…

sketch the graph of the following function. list the coordinates of where extrema or points of inflection occur. state where the function is increasing or decreasing as well as where it is concave up or concave down.\n\nf(x)=-x³ + 9x² - 52\n\nwhat are the coordinates of the relative extrema? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the coordinates of the relative extrema are \n(simplify your answer. type an ordered pair. type an exact answer using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there are no relative extrema\n\nwhat are the coordinates of the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the coordinates of the inflection points are \n(simplify your answer. type an ordered pair. type an exact answer using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there are no inflection points\n\non what interval(s) is f increasing or decreasing?

sketch the graph of the following function. list the coordinates of where extrema or points of inflection occur. state where the function is increasing or decreasing as well as where it is concave up or concave down.\n\nf(x)=-x³ + 9x² - 52\n\nwhat are the coordinates of the relative extrema? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the coordinates of the relative extrema are \n(simplify your answer. type an ordered pair. type an exact answer using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there are no relative extrema\n\nwhat are the coordinates of the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the coordinates of the inflection points are \n(simplify your answer. type an ordered pair. type an exact answer using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there are no inflection points\n\non what interval(s) is f increasing or decreasing?

Answer

Explanation:

Step1: Find the first - derivative

Given $f(x)=-x^{3}+9x^{2}-52$, then $f'(x)=-3x^{2}+18x=-3x(x - 6)$.

Step2: Find critical points

Set $f'(x)=0$, so $-3x(x - 6)=0$. Solving gives $x = 0$ and $x = 6$.

Step3: Determine increasing and decreasing intervals

Test intervals: For $x<0$, let $x=-1$, then $f'(-1)=-3\times(-1)\times(-1 - 6)=-21<0$, so $f(x)$ is decreasing on $(-\infty,0)$. For $0<x<6$, let $x = 1$, then $f'(1)=-3\times1\times(1 - 6)=15>0$, so $f(x)$ is increasing on $(0,6)$. For $x>6$, let $x = 7$, then $f'(7)=-3\times7\times(7 - 6)=-21<0$, so $f(x)$ is decreasing on $(6,\infty)$. Relative minima and maxima: Since $f(x)$ changes from decreasing to increasing at $x = 0$, $f(0)=-52$, so $(0,-52)$ is a relative minimum. Since $f(x)$ changes from increasing to decreasing at $x = 6$, $f(6)=-6^{3}+9\times6^{2}-52=-216 + 324-52=56$, so $(6,56)$ is a relative maximum.

Step4: Find the second - derivative

$f''(x)=-6x + 18$.

Step5: Find inflection points

Set $f''(x)=0$, so $-6x + 18=0$, which gives $x = 3$. And $f(3)=-3^{3}+9\times3^{2}-52=-27 + 81-52=2$. So the inflection point is $(3,2)$.

Answer:

What are the coordinates of the relative extrema? A. The coordinates of the relative extrema are $(0,-52),(6,56)$ What are the coordinates of the inflection points? A. The coordinates of the inflection points are $(3,2)$ On what interval(s) is $f$ increasing or decreasing? $f$ is decreasing on $(-\infty,0)\cup(6,\infty)$ and increasing on $(0,6)$