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}-9 x}{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 ( quad(x, y)=(quad) )\n\nrelative minimum ( quad(x, y)=(quad) )

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}-9 x}{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 ( quad(x, y)=(quad) )\n\nrelative minimum ( quad(x, y)=(quad) )

Answer

Explanation:

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

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 the 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)(x - 1)=0 ) gives ( x=-1,x = 1 ).

Step3: Determine the intervals of increase and decrease

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

Step4: Apply the First - Derivative Test

  • For ( x=-1 ):
    • ( f(-1)=\frac{(-1)^{9}-9\times(-1)}{9}=\frac{-1 + 9}{9}=\frac{8}{9} ). Since ( f^\prime(x) ) changes from positive (left of ( x=-1 )) to negative (right of ( x=-1 )), ( (-1,\frac{8}{9}) ) is a relative maximum.
  • For ( x = 1 ):
    • ( f(1)=\frac{1^{9}-9\times1}{9}=\frac{1 - 9}{9}=-\frac{8}{9} ). Since ( f^\prime(x) ) changes from negative (left of ( x = 1 )) to positive (right of ( x = 1 )), ( (1,-\frac{8}{9}) ) is a relative minimum.

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}) )