Question

c(x)=x^{1 / 5}(x + 6)
(a) find the interval(s) of increase. (enter your answer using interval notation.)
(-1,0),(0,\infty)
find the interval(s) of decrease. (enter your answer using interval notation.)
(-\infty, 1)
(b) find the local minimum value(s). (enter your answers as a comma - separated list.)
-5
find the local maximum value(s). (enter your answers as a comma - separated list.)
dne
(c) find the inflection points.
smaller x - value (x,y)=( - 2.297)
larger x - value (x,y)=(dne)
find the interval(s) where the function is concave up. (enter your answer using interval notation.)
(-\infty,-4)
find the interval(s) where the function is concave down. (enter your answer using interval notation.)
(-4,0),(0,\infty)

Explanation:

Step1: Find the first - derivative

Using the product rule \((uv)^\prime = u^\prime v+uv^\prime\), where \(u = x^{\frac{1}{5}}\) and \(v=x + 6\).
\(u^\prime=\frac{1}{5}x^{-\frac{4}{5}}\) and \(v^\prime = 1\).
So \(C^\prime(x)=\frac{1}{5}x^{-\frac{4}{5}}(x + 6)+x^{\frac{1}{5}}\times1=\frac{x + 6+5x}{5x^{\frac{4}{5}}}=\frac{6(x + 1)}{5x^{\frac{4}{5}}}\).
Set \(C^\prime(x)=0\), then \(x=-1\). The function \(C^\prime(x)\) is undefined at \(x = 0\).
Test intervals:

• For \(x<-1\), let \(x=-2\), \(C^\prime(-2)=\frac{6(-2 + 1)}{5(-2)^{\frac{4}{5}}}<0\).
• For \(-1
• For \(x>0\), let \(x = 1\), \(C^\prime(1)=\frac{6(1 + 1)}{5(1)^{\frac{4}{5}}}>0\).

Step2: Find the second - derivative

Using the quotient rule \((\frac{u}{v})^\prime=\frac{u^\prime v-uv^\prime}{v^{2}}\), where \(u = 6(x + 1)\) and \(v = 5x^{\frac{4}{5}}\).
\(u^\prime=6\) and \(v^\prime=4x^{-\frac{1}{5}}\).
\(C^{\prime\prime}(x)=\frac{6\times5x^{\frac{4}{5}}-6(x + 1)\times4x^{-\frac{1}{5}}}{25x^{\frac{8}{5}}}=\frac{30x-24(x + 1)}{25x^{\frac{9}{5}}}=\frac{6(x-4)}{25x^{\frac{9}{5}}}\).
Set \(C^{\prime\prime}(x)=0\), then \(x = 4\). The function \(C^{\prime\prime}(x)\) is undefined at \(x = 0\).
Test intervals:

• For \(x<0\), let \(x=-1\), \(C^{\prime\prime}(-1)=\frac{6(-1-4)}{25(-1)^{\frac{9}{5}}}>0\).
• For \(0
• For \(x>4\), let \(x = 5\), \(C^{\prime\prime}(5)=\frac{6(5 - 4)}{25(5)^{\frac{9}{5}}}>0\).

Answer:

(a) Interval of increase: \((-1,0)\cup(0,\infty)\)
Interval of decrease: \((-\infty,-1)\)
(b) Local minimum value: \(-5\)
Local maximum value: DNE
(c) Inflection points: \((4,\ 4^{\frac{1}{5}}(4 + 6))=(4,10\times4^{\frac{1}{5}})\approx(4,16.79)\)
Interval where the function is concave up: \((-\infty,0)\cup(4,\infty)\)
Interval where the function is concave down: \((0,4)\)