for the following function, a) give the coordinates of any critical points and classify each point as a…

for the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither, b) identify intervals where the function is increasing or decreasing, c) give the coordinates of any points of inflection, d) identify intervals where the function is concave up or concave down, and e) sketch the graph. h(x)=3x^3 - 9x\na the relative minimum point(s) is/are (1, - 6) and the relative maximum point(s) is/are (-1,6) (simplify your answers. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.)\nb the relative minimum point(s) is/are and there are no relative maximum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.)\nc the relative maximum point(s) is/are and there are no relative minimum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.)\nd there are no relative minimum points and there are no relative maximum points\nb) on what interval(s) is h increasing or decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is increasing on the function is decreasing on (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.)\nb. the function is decreasing on the function is never increasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)\nc. the function is increasing on the function is never decreasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)\nd. the function is never increasing or decreasing

for the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither, b) identify intervals where the function is increasing or decreasing, c) give the coordinates of any points of inflection, d) identify intervals where the function is concave up or concave down, and e) sketch the graph. h(x)=3x^3 - 9x\na the relative minimum point(s) is/are (1, - 6) and the relative maximum point(s) is/are (-1,6) (simplify your answers. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.)\nb the relative minimum point(s) is/are and there are no relative maximum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.)\nc the relative maximum point(s) is/are and there are no relative minimum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.)\nd there are no relative minimum points and there are no relative maximum points\nb) on what interval(s) is h increasing or decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. the function is increasing on the function is decreasing on (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.)\nb. the function is decreasing on the function is never increasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)\nc. the function is increasing on the function is never decreasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)\nd. the function is never increasing or decreasing

Answer

Explanation:

Step1: Find the first - derivative

Differentiate $h(x)=3x^{3}-9x$ using the power rule. If $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. So $h^\prime(x)=9x^{2}-9 = 9(x^{2}-1)=9(x + 1)(x - 1)$.

Step2: Find critical points

Set $h^\prime(x)=0$. Then $9(x + 1)(x - 1)=0$. Solving for $x$, we get $x=-1$ or $x = 1$. Substitute $x=-1$ into $h(x)$: $h(-1)=3(-1)^{3}-9(-1)=-3 + 9=6$. Substitute $x = 1$ into $h(x)$: $h(1)=3(1)^{3}-9(1)=3-9=-6$. So the critical points are $(-1,6)$ and $(1,-6)$.

Step3: Use the first - derivative test for increasing and decreasing intervals

Choose test points in the intervals $(-\infty,-1)$, $(-1,1)$ and $(1,\infty)$. For the interval $(-\infty,-1)$, let $x=-2$. Then $h^\prime(-2)=9((-2)^{2}-1)=9(4 - 1)=27>0$, so $h(x)$ is increasing on $(-\infty,-1)$. For the interval $(-1,1)$, let $x = 0$. Then $h^\prime(0)=9(0^{2}-1)=-9<0$, so $h(x)$ is decreasing on $(-1,1)$. For the interval $(1,\infty)$, let $x = 2$. Then $h^\prime(2)=9(2^{2}-1)=9(4 - 1)=27>0$, so $h(x)$ is increasing on $(1,\infty)$.

  • a) The relative minimum point is $(1,-6)$ and the relative maximum point is $(-1,6)$. So the answer for part a) is A. The relative minimum point(s) is/are $(1,-6)$ and the relative maximum point(s) is/are $(-1,6)$.
  • b) The function is increasing on $(-\infty,-1)\cup(1,\infty)$ and decreasing on $(-1,1)$. So the answer for part b) is A. The function is increasing on $(-\infty,-1),(1,\infty)$; The function is decreasing on $(-1,1)$.

Answer:

a) A. The relative minimum point(s) is/are $(1,-6)$ and the relative maximum point(s) is/are $(-1,6)$ b) A. The function is increasing on $(-\infty,-1),(1,\infty)$; The function is decreasing on $(-1,1)$