Question

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

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

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.

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

a. the absolute maximum point(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 absolute maximum points.

Explanation:

Step1: Find the first derivative

Use the power rule \((x^n)^\prime = nx^{n - 1}\).
For \(y = 12x^{3}-x^{4}\), \(y^\prime=36x^{2}-4x^{3}=4x^{2}(9 - x)\)
Set \(y^\prime = 0\), then \(4x^{2}(9 - x)=0\).
Solutions are \(x = 0\) and \(x = 9\).

Step2: Use the second - derivative test

Find the second derivative. \(y^{\prime\prime}=72x-12x^{2}=12x(6 - x)\)

• When \(x = 0\): \(y^{\prime\prime}(0)=0\), the second - derivative test is inconclusive.
• When \(x = 9\): \(y^{\prime\prime}(9)=72\times9-12\times9^{2}=648 - 972=-324<0\)

So \(x = 9\) is a local maximum. \(y(9)=12\times9^{3}-9^{4}=12\times729 - 6561=8748 - 6561 = 2187\)

Step3: Analyze the end - behavior

As \(x
ightarrow\pm\infty\), \(y = 12x^{3}-x^{4}=-x^{4}(1-\frac{12}{x})
ightarrow-\infty\)
Since \(y(9) = 2187\) and \(y
ightarrow-\infty\) as \(x
ightarrow\pm\infty\), the function has an absolute maximum at \((9,2187)\)

Answer:

For the local minimum: B. There are no local minimum points
For the absolute maximum: A. The absolute maximum point(s) is/are \((9,2187)\)