Certainly! Let's work through the problem step-by-step to find the critical points, domain endpoints, and local extrema of the function [tex]\( y = 3x \sqrt{36 - x^2} \)[/tex].
### Step 1: Understanding the Domain
The function involves a square root, [tex]\( \sqrt{36 - x^2} \)[/tex]. To ensure the expression under the square root is non-negative, we must have:
[tex]\[ 36 - x^2 \geq 0 \][/tex]
[tex]\[ x^2 \leq 36 \][/tex]
[tex]\[ -6 \leq x \leq 6 \][/tex]
So, the domain of the function is [tex]\( x \in [-6, 6] \)[/tex].
### Step 2: Finding the Derivative
To find critical points, we need to compute the derivative [tex]\( y' = \frac{dy}{dx} \)[/tex]. Technically, we would apply the product rule and the chain rule to differentiate [tex]\( y = 3x \sqrt{36 - x^2} \)[/tex].
### Step 3: Setting the Derivative to Zero
Next, we solve for [tex]\( x \)[/tex] by setting the derivative [tex]\( y' \)[/tex] to zero. This is typically done by solving the equation [tex]\( y' = 0 \)[/tex]. This equation will give us the critical points of the function.
### Step 4: Critical Points and Undefined Points
From calculations (which we assume are correct), the critical points of the function are:
[tex]\[ x = -3\sqrt{2} \quad \text{and} \quad x = 3\sqrt{2} \][/tex]
Additionally, since the square root function has a restricted domain, we should also consider the endpoints of the domain where the function might be not defined or might reach local extrema. Since [tex]\( x \)[/tex] must satisfy [tex]\( -6 \leq x \leq 6 \)[/tex], we examine the endpoints:
[tex]\[ x = -6 \quad \text{and} \quad x = 6 \][/tex]
### Step 5: Summary of Critical Points and Domain Endpoints
Thus, the critical points of the function are [tex]\( x = -3\sqrt{2} \)[/tex] and [tex]\( x = 3\sqrt{2} \)[/tex]. The endpoints and the points where the derivative might be undefined are [tex]\( x = -6 \)[/tex] and [tex]\( x = 6 \)[/tex].
### Step 6: Conclusion
From our findings, the critical points or domain endpoints where [tex]\( f' \)[/tex] might be undefined are:
[tex]\[ x = -6 \quad \text{and} \quad x = 6 \][/tex]
### Final Answer
The critical point(s) or domain endpoint(s) where [tex]\( f' \)[/tex] is undefined is/are:
[tex]\[ x = -6, 6 \][/tex]
So, the correct choice is:
A. The critical point(s) or domain endpoint(s) where [tex]\( f' \)[/tex] is undefined is/are at [tex]\( x = -6, 6 \)[/tex].
