summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=4 x^{6}-9 x^{5} )\nthe ( x )-intercept(s) of ( f ) is/are ( x = 0,\frac{9}{4} ).\n(type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed)\nthe function ( f ) has no ( x )-intercepts.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\nthe ( y )-intercept of ( f ) is ( y = 0 ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nthe function ( f ) has no ( y )-intercept.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the function ( f ) is increasing on the subinterval(s)\n(type your answer in interval notation. type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed.)\nb. the function ( f ) is never increasing.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=4 x^{6}-9 x^{5} )\nthe ( x )-intercept(s) of ( f ) is/are ( x = 0,\frac{9}{4} ).\n(type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed)\nthe function ( f ) has no ( x )-intercepts.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\nthe ( y )-intercept of ( f ) is ( y = 0 ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nthe function ( f ) has no ( y )-intercept.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the function ( f ) is increasing on the subinterval(s)\n(type your answer in interval notation. type an exact answer, using radicals as needed. type an integer or a fraction. use a comma to separate answers as needed.)\nb. the function ( f ) is never increasing.

Answer

Explanation:

Step1: Find the derivative

First, find the derivative of (y = f(x)=4x^{6}-9x^{5}). Using the power rule ((x^{n})^\prime=nx^{n - 1}), we have (f^\prime(x)=24x^{5}-45x^{4}=3x^{4}(8x - 15)).

Step2: Determine critical points

Set (f^\prime(x)=0). Since (3x^{4}(8x - 15)=0), then (x = 0) or (x=\frac{15}{8}).

Step3: Test intervals for increasing/decreasing

  • For (x<0), let (x=-1). Then (f^\prime(-1)=3(-1)^{4}[8(-1)-15]=3\times(-23)<0).
  • For (0<x<\frac{15}{8}), let (x = 1). Then (f^\prime(1)=3(1)^{4}(8\times1 - 15)=3\times(-7)<0).
  • For (x>\frac{15}{8}), let (x = 2). Then (f^\prime(2)=3(2)^{4}(8\times2 - 15)=3\times16\times1>0).

Answer:

A. The function (f) is increasing on the sub - interval ((\frac{15}{8},\infty))