consider the following function. use a graphing utility to confirm your answers for parts (a) through (c)…

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.)\n\n$f(x)=\\frac{x^{9}-9x}{9}$\n\n(a) find the critical numbers of $f$. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=()$\n\nrelative minimum $(x,y)=()$

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.)\n\n$f(x)=\\frac{x^{9}-9x}{9}$\n\n(a) find the critical numbers of $f$. (enter your answers as a comma-separated list.)\n\n$x=$\n\n(b) find the open intervals on which the function is increasing or decreasing. (enter your answers using interval notation.)\n\nincreasing\n\ndecreasing\n\n(c) apply the first derivative test to identify all relative extrema.\n\nrelative maximum $(x,y)=()$\n\nrelative minimum $(x,y)=()$

Answer

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<x<1 ), let ( x = 0 ). Then ( f^\prime(0)=0^{8}-1=-1<0 ).
    • 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<x<1 ), ( f^\prime(x)<0 )), then ( f(-1)=\frac{(-1)^{9}-9\times(-1)}{9}=\frac{-1 + 9}{9}=\frac{8}{9} ). So the relative maximum is at ( (-1,\frac{8}{9}) ).
    • At ( x = 1 ):
      • Since ( f(x) ) changes from decreasing (( -1<x<1 ), ( f^\prime(x)<0 )) to increasing (( x>1 ), ( 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}) )