Question

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

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
• 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: \((-\infty,4)\)