Question

consider the following. (if an answer does not exist, enter dne.)
$f(x)=x^{3}+3 x^{2}-72 x$
(a) find the interval(s) on which $f$ is increasing. (enter your answer using interval notation.)

(b) find the interval(s) on which $f$ is decreasing. (enter your answer using interval notation.)

(c) find the local minimum and maximum value of $f$.
local minimum value

local maximum value

Explanation:

Step1: Find the derivative of \(f(x)\)

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(f(x)=x^{3}+3x^{2}-72x\), we have \(f^\prime(x)=3x^{2}+6x - 72\). Factor out a \(3\): \(f^\prime(x)=3(x^{2}+2x - 24)\). Then factor the quadratic: \(f^\prime(x)=3(x + 6)(x - 4)\).

Step2: Find the critical points

Set \(f^\prime(x)=0\). So \(3(x + 6)(x - 4)=0\). Using the zero - product property \(x+6 = 0\) or \(x - 4=0\). The critical points are \(x=-6\) and \(x = 4\).

Step3: Determine the sign of \(f^\prime(x)\) in the intervals

We consider the intervals \((-\infty,-6)\), \((-6,4)\), and \((4,\infty)\).

• For \(x\in(-\infty,-6)\), let \(x=-7\). Then \(f^\prime(-7)=3(-7 + 6)(-7 - 4)=3(-1)(-11)=33>0\).
• For \(x\in(-6,4)\), let \(x = 0\). Then \(f^\prime(0)=3(0 + 6)(0 - 4)=3\times6\times(-4)=-72<0\).
• For \(x\in(4,\infty)\), let \(x = 5\). Then \(f^\prime(5)=3(5 + 6)(5 - 4)=3\times11\times1 = 33>0\).

Step4: Find the intervals of increase and decrease

Since \(f^\prime(x)>0\) on \((-\infty,-6)\cup(4,\infty)\), the function \(f(x)\) is increasing on \((-\infty,-6)\cup(4,\infty)\).
Since \(f^\prime(x)<0\) on \((-6,4)\), the function \(f(x)\) is decreasing on \((-6,4)\).

Step5: Find the local minimum and maximum values

We use the second - derivative test. The second derivative \(f^{\prime\prime}(x)=6x + 6\).

• For \(x=-6\), \(f^{\prime\prime}(-6)=6\times(-6)+6=-30<0\). So \(f(-6)=(-6)^{3}+3(-6)^{2}-72(-6)=-216 + 108+432=324\) is a local maximum.
• For \(x = 4\), \(f^{\prime\prime}(4)=6\times4+6=30>0\). So \(f(4)=4^{3}+3\times4^{2}-72\times4=64 + 48-288=-176\) is a local minimum.

Answer:

(a) \((-\infty,-6)\cup(4,\infty)\)
(b) \((-6,4)\)
(c) local minimum value: \(-176\), local maximum value: \(324\)