Question

consider the following function. use a graphing utility to confirm your answers for parts (a) through (c). (if an answer does not exist, enter dne.)

$f(x)=\frac{x^{9}-9x}{9}$

(a) find the critical numbers of $f$. (enter your answers as a comma-separated list.)

$x=$

(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)

increasing

decreasing

(c) apply the first derivative test to identify all relative extrema.

relative maximum $(x,y)=()$

relative minimum $(x,y)=()$

Explanation:

Step1: Find the derivative

Given \( f(x)=\frac{x^{9}-9x}{9}=\frac{1}{9}x^{9}-x \).
Using the power rule \( (x^n)^\prime = nx^{n - 1} \), the derivative \( f^\prime(x)=\frac{1}{9}\times9x^{8}-1=x^{8}-1=(x^{4}+1)(x^{4}-1)=(x^{4}+1)(x^{2}+1)(x + 1)(x - 1) \).

Step2: Find critical numbers

Set \( f^\prime(x)=0 \).
Since \( x^{4}+1>0 \) and \( x^{2}+1>0 \) for all real \( x \), then \( (x + 1)(x - 1)=0 \).
Solving \( x+1 = 0 \) gives \( x=-1 \), and solving \( x - 1=0 \) gives \( x = 1 \).

Step3: Determine intervals of increase and decrease

• Test intervals:
• For \( x<-1 \), let \( x=-2 \). Then \( f^\prime(-2)=(-2)^{8}-1=256 - 1=255>0 \).
• For \( -1
• For \( x>1 \), let \( x = 2 \). Then \( f^\prime(2)=2^{8}-1=256 - 1=255>0 \).
• Intervals:
• The function is increasing when \( f^\prime(x)>0 \), so the increasing intervals are \( (-\infty,-1)\cup(1,\infty) \).
• The function is decreasing when \( f^\prime(x)<0 \), so the decreasing interval is \( (-1,1) \).

Step4: Apply the First - Derivative Test

• Relative extrema:
• At \( x=-1 \):
• Since \( f(x) \) changes from increasing (\( x<-1 \), \( f^\prime(x)>0 \)) to decreasing (\( -1
• At \( x = 1 \):
• Since \( f(x) \) changes from decreasing (\( -11 \), \( f^\prime(x)>0 \)), then \( f(1)=\frac{1^{9}-9\times1}{9}=\frac{1 - 9}{9}=-\frac{8}{9} \). So the relative minimum is at \( (1,-\frac{8}{9}) \).

Answer:

(a) \( x=-1,1 \)
(b) Increasing: \( (-\infty,-1)\cup(1,\infty) \); Decreasing: \( (-1,1) \)
(c) Relative maximum: \( (-1,\frac{8}{9}) \); Relative minimum: \( (1,-\frac{8}{9}) \)