 yequalsRootIndex 3 StartRoot 2 x cubed plus 16 EndRoot entify where the curve is increasing or decreasing. Select the correct choice below​ and, if​ necessary, fill in the answer​ box(es) to complete your choice.
A.
The curve increases on the open​ interval(s) 
  
left parenthesis negative 2 comma 0 right parenthesis comma left parenthesis 0 comma infinity right parenthesis and decreases on the open​ interval(s) 
  
left parenthesis negative infinity comma negative 2 right parenthesis.
​(Type your answers in interval notation. Use a comma to separate answers as​ needed.)
B.
The curve increases on the open​ interval(s) 
  
left parenthesis negative infinity comma negative 2 right parenthesis comma left parenthesis negative 2 comma 0 right parenthesis comma left parenthesis 0 comma infinity right parenthesis and does not decrease.
​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)
C.
The curve does not increase and decreases on the open​ interval(s) 
  
enter your response here.
​(Type your answer in interval notation. Use a comma to separate answers as​ needed.)
D.
The curve neither increases nor decreases. Suppose f is continuous on​ [a,b] and differentiable on​ (a,b). If f prime​(x)greater than0 at each point xis an element of​(a,b), then f is increasing on​ [a,b]. Similarly, If f prime​(x)less than0 at each point xis an element of​(a,b), then f is decreasing on​ [a,b].


We are given the function \( y = \sqrt[3]{2x^3 + 16} \) and need to determine where the curve is increasing or decreasing. To do that, we need to compute the first derivative \( y' \) of the function and analyze its sign.

### Step 1: Differentiate the function

We have the function:
\[
y = (2x^3 + 16)^{1/3}
\]
To find where the function is increasing or decreasing, we need to compute the derivative using the chain rule:

\[
y' = \frac{d}{dx} \left( (2x^3 + 16)^{1/3} \right)
\]

Using the chain rule:
\[
y' = \frac{1}{3}(2x^3 + 16)^{-2/3} \cdot \frac{d}{dx}(2x^3 + 16)
\]

Now, differentiate the inside function \( 2x^3 + 16 \):
\[
\frac{d}{dx}(2x^3 + 16) = 6x^2
\]

So the derivative is:
\[
y' = \frac{1}{3}(2x^3 + 16)^{-2/3} \cdot 6x^2
\]
\[
y' = \frac{2x^2}{(2x^3 + 16)^{2/3}}
\]

### Step 2: Analyze the derivative

Now we need to determine where \( y' \) is positive, negative, or zero.

- **Critical points** occur when \( y' = 0 \) or when the derivative is undefined. Set \( y' = 0 \) to find critical points:
\[
\frac{2x^2}{(2x^3 + 16)^{2/3}} = 0
\]
This occurs when \( x^2 = 0 \), which gives \( x = 0 \).

Next, check where the denominator \( (2x^3 + 16)^{2/3} \) is undefined. It is always positive since \( 2x^3 + 16 \geq 0 \) for all real \( x \). Hence, the derivative does not become undefined for any real value of \( x \).

### Step 3: Determine intervals of increasing and decreasing

To classify the intervals, check the sign of \( y' \) in different intervals around the critical point \( x = 0 \).

- For \( x > 0 \) (positive values of \( x \)), the derivative \( y' = \frac{2x^2}{(2x^3 + 16)^{2/3}} \) is positive because both the numerator and denominator are positive. Hence, the function is **increasing** for \( x > 0 \).
- For \( x < 0 \), check if \( y' \) is positive or negative. Since \( x^2 \) is positive and \( 2x^3 + 16 \) is still positive for \( x < 0 \), the derivative \( y' \) is positive in this interval as well.

Therefore, the function is **increasing** for all \( x \), and there is no interval where the function is decreasing.

### Conclusion

The correct answer is **B**:  
The curve increases on the open intervals \( (-\infty, -2) \), \( (-2, 0) \), and \( (0, \infty) \), and does not decrease.

---

Would you like any further clarifications or details on these steps?

Here are 5 related questions for deeper understanding:
1. How do you determine where a function is increasing or decreasing?
2. What is the role of the first derivative in analyzing function behavior?
3. Why does the function not decrease for any value of \( x \)?
4. How would you analyze the concavity of this function?
5. Can a function increase or decrease on a closed interval?

**Tip:** Always check both the numerator and denominator when analyzing the sign of a fraction derivative.

Identify where the curve y = RootIndex 3 StartRoot 2x^3 + 16 EndRoot is increasing or decreasing. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice: A) The curve increases on the open intervals (-2, 0) and (0, infinity) and decreases on (-infinity, -2). B) The curve increases on the open intervals (-infinity, -2), (-2, 0), and (0, infinity) and does not decrease. C) The curve does not increase and decreases on the open interval(s). D) The curve neither increases nor decreases.

This calculus problem involves determining where the function y = (2x^3 + 16)^(1/3) is increasing or decreasing. Using the first derivative test, we find the function is increasing on intervals (-infinity, -2), (-2, 0), and (0, infinity). This solution is ideal for students in Grades 10-12.

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