Question

identify the coordinates of any local and absolute extreme points and inflection points. graph the function.
y = 12x³ - x⁴ = x³(12 - x)
if necessary, fill in the answer box to complete your choice.
a. the absolute 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 absolute maximum points.
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.

Explanation:

Step1: Analyze the function as \(x\to\pm\infty\)

As \(x\to\infty\), \(y = 12x^{3}-x^{4}=x^{3}(12 - x)\approx - x^{4}\to-\infty\). As \(x\to-\infty\), \(y = 12x^{3}-x^{4}=x^{3}(12 - x)\approx12x^{3}\to-\infty\).

Step2: Conclusion about absolute minimum

Since \(\lim_{x\to\pm\infty}y =-\infty\), the function \(y = 12x^{3}-x^{4}\) has no lower - bound.

Answer:

B. There are no absolute minimum points.