consider the equation below. (if an answer does not exist, enter dne.)\n$f(x)=x^{3}-12x^{2}-27x + 7$\n(a)…

consider the equation below. (if an answer does not exist, enter dne.)\n$f(x)=x^{3}-12x^{2}-27x + 7$\n(a) find the interval on which $f$ is increasing. (enter your answer using interval notation.)\nfind the interval on which $f$ is decreasing. (enter your answer using interval notation.)\n(b) find the local minimum and maximum values of $f$.\nlocal minimum value\nlocal maximum value\n(c) find the inflection point.\n$(x,y)=$( )\nfind the interval on which $f$ is concave up. (enter your answer using interval notation.)\nfind the interval on which $f$ is concave down. (enter your answer using interval notation.)

consider the equation below. (if an answer does not exist, enter dne.)\n$f(x)=x^{3}-12x^{2}-27x + 7$\n(a) find the interval on which $f$ is increasing. (enter your answer using interval notation.)\nfind the interval on which $f$ is decreasing. (enter your answer using interval notation.)\n(b) find the local minimum and maximum values of $f$.\nlocal minimum value\nlocal maximum value\n(c) find the inflection point.\n$(x,y)=$( )\nfind the interval on which $f$ is concave up. (enter your answer using interval notation.)\nfind the interval on which $f$ is concave down. (enter your answer using interval notation.)

Answer

Explanation:

Step1: Find the first derivative

The derivative of (f(x)=x^{3}-12x^{2}-27x + 7) is (f^\prime(x)=3x^{2}-24x - 27=3(x^{2}-8x - 9)=3(x + 1)(x - 9)).

Step2: Determine critical points

Set (f^\prime(x)=0), then (3(x + 1)(x - 9)=0). So (x=-1) and (x = 9).

Step3: Analyze the sign of (f^\prime(x))

  • For (x<-1), let (x=-2), (f^\prime(-2)=3(-2 + 1)(-2 - 9)=3\times(-1)\times(-11)=33>0).
  • For (-1<x<9), let (x = 0), (f^\prime(0)=3(0 + 1)(0 - 9)=-27<0).
  • For (x>9), let (x = 10), (f^\prime(10)=3(10 + 1)(10 - 9)=33>0).

So (f(x)) is increasing on ((-\infty,-1)\cup(9,\infty)) and decreasing on ((-1,9)).

Step4: Find local extrema

  • (f(-1)=(-1)^{3}-12(-1)^{2}-27(-1)+7=-1-12 + 27+7=21).
  • (f(9)=9^{3}-12\times9^{2}-27\times9+7=729-972-243 + 7=-479).

So the local maximum value is (21) and the local minimum value is (-479).

Step5: Find the second derivative

(f^{\prime\prime}(x)=6x-24=6(x - 4)).

Step6: Determine inflection point

Set (f^{\prime\prime}(x)=0), then (x = 4). (f(4)=4^{3}-12\times4^{2}-27\times4+7=64-192-108 + 7=-229). So the inflection point is ((4,-229)).

Step7: Analyze the sign of (f^{\prime\prime}(x))

  • For (x<4), let (x = 0), (f^{\prime\prime}(0)=6\times0-24=-24<0).
  • For (x>4), let (x = 5), (f^{\prime\prime}(5)=6\times5-24=6>0).

So (f(x)) is concave up on ((4,\infty)) and concave down on ((-\infty,4)).

Answer:

(a) Increasing: ((-\infty,-1)\cup(9,\infty)); Decreasing: ((-1,9)) (b) Local minimum value: (-479); Local maximum value: (21) (c) Inflection point: ((4,-229)); Concave up: ((4,\infty)); Concave down: ((-\infty,4))