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 ( 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.\nthe function ( f ) is increasing on the subinterval(s) ( left(\frac{15}{8}, infty\right) )\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)\nthe function ( f ) is never increasing\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the function ( f ) is decreasing 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 decreasing

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 ( 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.\nthe function ( f ) is increasing on the subinterval(s) ( left(\frac{15}{8}, infty\right) )\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)\nthe function ( f ) is never increasing\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the function ( f ) is decreasing 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 decreasing

Answer

Explanation:

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

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (f(x)=4x^{6}-9x^{5}), the derivative (f^\prime(x)=24x^{5}-45x^{4}=3x^{4}(8x - 15))

Step2: Determine the critical points

Set (f^\prime(x)=0). Since (3x^{4}(8x - 15)=0), we have (x = 0) (from (x^{4}=0)) and (x=\frac{15}{8}) (from (8x-15 = 0))

Step3: Analyze the sign of (f^\prime(x))

  • For (x<\frac{15}{8}) and (x\neq0), let's take a test - point (x = 1). Then (f^\prime(1)=24\times1^{5}-45\times1^{4}=24 - 45=-21<0)
  • For (x>\frac{15}{8}), let's take a test - point (x = 2). Then (f^\prime(2)=24\times2^{5}-45\times2^{4}=24\times32-45\times16=768 - 720 = 48>0)

Answer:

The function (f(x)) is decreasing on the sub - interval (\left(-\infty,\frac{15}{8}\right))