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 function ( f ) is increasing on the subinterval(s) ( left(\frac{-infty}{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\nthe function ( f ) is decreasing on the subinterval(s) ( left(-infty, \frac{15}{8}\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 decreasing\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the function ( f ) has a local maximum at ( x )\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no local maxima

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 function ( f ) is increasing on the subinterval(s) ( left(\frac{-infty}{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\nthe function ( f ) is decreasing on the subinterval(s) ( left(-infty, \frac{15}{8}\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 decreasing\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice\na. the function ( f ) has a local maximum at ( x )\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. the function ( f ) has no local maxima

Answer

Explanation:

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

Use the power rule ((x^n)^\prime = nx^{n - 1}). (f^\prime(x)=(4x^{6}-9x^{5})^\prime=4\times6x^{5}-9\times5x^{4}=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: 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)=3\times1^{4}(8\times1 - 15)=3\times(- 7)<0)
  • For (x>\frac{15}{8}), let's take a test - point (x = 2). Then (f^\prime(2)=3\times2^{4}(8\times2 - 15)=3\times16\times1>0)

Since the function changes from decreasing to increasing at (x=\frac{15}{8}) and (f^\prime(x)) does not change sign at (x = 0) (because (x^{4}\geqslant0) for all (x) and the sign of (f^\prime(x)) near (x = 0) is determined by (8x-15)).

Answer:

The function (f) has no local maxima.