Question

find the critical point(s) within the domain of the following function. answer exactly.

\\y = 16x^3 - 12x\\

\\x = \\

Explanation:

Identify the function and domain

The given function is:

$$ y = 16x^3 - 12x $$

Since this is a polynomial function, its domain is all real numbers:

$$ x \in (-\infty, \infty) $$

Find the first derivative

Using the Derivative Test knowledge point

$$ y' = \frac{d}{dx}(16x^3 - 12x) = 48x^2 - 12 $$

Set the derivative to zero

Using the Derivative Test knowledge point

$$
LATEXBLOCK0
$$

Solve for the critical points

Using the Derivative Test knowledge point

$$ x = \pm\sqrt{\frac{1}{4}} = \pm\frac{1}{2} $$

Both values lie within the domain of the function.

Answer:

Find the critical point(s) within the domain of the following function. Answer exactly.
\(y = 16x^3 - 12x\)
\(x =\) <blank>\(\pm\frac{1}{2}\)</blank>