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 absolute minimum points. select the correct choice below and,
if necessary, fill in the answer box to complete your choice.

a. the absolute minimum 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 minimum points.

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

a. the inflection 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 inflection points.

Explanation:

Step1: Find the first derivative

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

Step2: Find the second derivative

Differentiate \(y^\prime = 36x^2-4x^3\) using the power rule.
\(y^{\prime\prime}=72x-12x^2 = 12x(6 - x)\)

Step3: Find inflection points

Set \(y^{\prime\prime}=0\), so \(12x(6 - x)=0\)
Solving \(12x(6 - x)=0\) gives \(x = 0\) or \(x = 6\)
When \(x = 0\), \(y=12\times0^3-0^4 = 0\)
When \(x = 6\), \(y=12\times6^3-6^4=12\times216 - 1296=2592-1296 = 1296\)

Answer:

A. The inflection point(s) is/are \((0,0),(6,1296)\)