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

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 \(x=\frac{15}{8}\)