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 function ( f ) is increasing on the subinterval(s) ( left(\frac{-infty}{8}, infty
ight) ).
(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.)
the function ( f ) is never increasing.
select the correct choice below and, if necessary, fill in the answer box to complete your choice
the function ( f ) is decreasing on the subinterval(s) ( left(-infty, \frac{15}{8}
ight) )
(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.)
the function ( f ) is never decreasing
select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. the function ( f ) has a local maximum at ( x )
(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)
b. the function ( f ) has no local maxima

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

eq0\), 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.