Question

identify the coordinates of any local and absolute extreme points and inflection points. graph the function.

$y = 12x^{3}-x^{4}=x^{3}(12 - x)$

identify the coordinates of the local maximum points. select the correct choice below and,
if necessary, fill in the answer box to complete your choice.

a. the local maximum point(s) is/are $(9,2187)$
(use a comma to separate answers as needed. type an ordered pair. do not use commas in
the individual coordinates.)

b. there are no local maximum points.

identify the coordinates of the local minimum points. select the correct choice below and,
if necessary, fill in the answer box to complete your choice.

a. the local minimum points(s) is/are
(use a comma to separate answers as needed. type an ordered pair. do not use commas in
the individual coordinates.)

b. there are no local minimum points.

Explanation:

Step1: Find the first derivative

Given \(y = 12x^{3}-x^{4}\), using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(y^\prime=36x^{2}-4x^{3}=4x^{2}(9 - x)\).
Set \(y^\prime = 0\), then \(4x^{2}(9 - x)=0\). Solving this equation gives \(x = 0\) or \(x = 9\).

Step2: Use the first - derivative test

• For \(x<9\) (say \(x = 1\)), \(y^\prime=4\times1^{2}(9 - 1)=32>0\).
• For \(x>9\) (say \(x = 10\)), \(y^\prime=4\times10^{2}(9 - 10)=- 400<0\).
• When \(x = 0\), in the neighborhood of \(x = 0\) (e.g., \(x=-1\), \(y^\prime=4\times(-1)^{2}(9+1) = 40>0\) and \(x = 1\), \(y^\prime=32>0\)), the sign of \(y^\prime\) does not change.

Step3: Find the second derivative

\(y^\prime=36x^{2}-4x^{3}\), then \(y^{\prime\prime}=72x-12x^{2}=12x(6 - x)\).
Set \(y^{\prime\prime}=0\), we get \(x = 0\) or \(x = 6\).

• When \(x = 0\), \(y(0)=0\).
• When \(x = 6\), \(y(6)=12\times6^{3}-6^{4}=12\times216-1296=2592 - 1296 = 1296\).

Answer:

For local minimum points:
Since the function does not change from decreasing to increasing at any critical point (except the behavior at \(x = 0\) where there is no minimum - like sign change of the first derivative), the answer is \(B\). There are no local minimum points.