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 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.\nthe 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.)\nthe function ( f ) has no local maximum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local minimum at ( x= )\n(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 ) has no local minimum

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 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.\nthe 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.)\nthe function ( f ) has no local maximum.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function ( f ) has a local minimum at ( x= )\n(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 ) has no local minimum

Answer

Explanation:

Step1: Find the first - derivative

Given (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 the critical points

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

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

  • For (x\lt\frac{15}{8}) (except (x = 0)), if we take a test point (x = 1), (f^\prime(1)=3\times1^{4}(8\times1 - 15)=3\times(- 7)\lt0).
  • For (x\gt\frac{15}{8}), if we take a test point (x = 2), (f^\prime(2)=3\times2^{4}(8\times2 - 15)=3\times16\times1\gt0).

Since the function changes from decreasing ((f^\prime(x)\lt0)) to increasing ((f^\prime(x)\gt0)) at (x=\frac{15}{8}) and the derivative (f^\prime(x)) does not change sign at (x = 0) (because of the even - multiplicity of the root (x = 0) for (f^\prime(x))).

Answer:

The function (f) has a local minimum at (x=\frac{15}{8})