Question

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

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
• 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 \((\frac{15}{8},\infty)\)