Describe intervaks for which the function is increasing, decreasing, or constant. 6. f(x) = 2x^2 7. f(x) = -3x^-5 8. g(x) = -x^3 ÷2

Let's analyze each function step by step to determine the intervals where the function is increasing, decreasing, or constant.

### 6. \( f(x) = 2x^2 \)
This is a quadratic function, which has the general form \( ax^2 \). The graph of this function is a parabola that opens upwards since the coefficient of \( x^2 \) is positive (\( a = 2 \)).

- **Critical points**: To find intervals of increase or decrease, we take the derivative \( f'(x) \):
  \[
  f'(x) = \frac{d}{dx} (2x^2) = 4x
  \]
- **Setting the derivative equal to 0** to find critical points:
  \[
  4x = 0 \quad \Rightarrow \quad x = 0
  \]
- **Behavior on intervals**:
  - For \( x < 0 \), \( f'(x) = 4x < 0 \), so the function is **decreasing**.
  - For \( x > 0 \), \( f'(x) = 4x > 0 \), so the function is **increasing**.

Thus, \( f(x) = 2x^2 \) is:
- **Decreasing** on \( (-\infty, 0) \)
- **Increasing** on \( (0, \infty) \)

### 7. \( f(x) = -3x^{-5} \)
This is a rational function involving a negative exponent.

- **Critical points**: Take the derivative \( f'(x) \):
  \[
  f'(x) = \frac{d}{dx} \left(-3x^{-5}\right) = 15x^{-6}
  \]
  or \( f'(x) = \frac{15}{x^6} \).
- Notice that \( x^6 > 0 \) for all \( x \neq 0 \), meaning that \( f'(x) > 0 \) for all \( x \neq 0 \). This means the function is always **increasing** except at \( x = 0 \), where the function is undefined (vertical asymptote).
  
Thus, \( f(x) = -3x^{-5} \) is:
- **Increasing** on \( (-\infty, 0) \cup (0, \infty) \)

### 8. \( g(x) = -\frac{x^3}{2} \)
This is a cubic function scaled by a constant factor.

- **Critical points**: Take the derivative \( g'(x) \):
  \[
  g'(x) = \frac{d}{dx} \left(-\frac{x^3}{2}\right) = -\frac{3x^2}{2}
  \]
- **Setting the derivative equal to 0** to find critical points:
  \[
  -\frac{3x^2}{2} = 0 \quad \Rightarrow \quad x = 0
  \]
- **Behavior on intervals**:
  - For \( x < 0 \), \( g'(x) = -\frac{3x^2}{2} < 0 \), so the function is **decreasing**.
  - For \( x > 0 \), \( g'(x) = -\frac{3x^2}{2} < 0 \), so the function is still **decreasing**.

Thus, \( g(x) = -\frac{x^3}{2} \) is:
- **Decreasing** on \( (-\infty, \infty) \)

### Summary:
1. \( f(x) = 2x^2 \)
   - Decreasing on \( (-\infty, 0) \)
   - Increasing on \( (0, \infty) \)
2. \( f(x) = -3x^{-5} \)
   - Increasing on \( (-\infty, 0) \cup (0, \infty) \)
3. \( g(x) = -\frac{x^3}{2} \)
   - Decreasing on \( (-\infty, \infty) \)

Would you like further details or clarifications on any of these functions?

### Related Questions:
1. How do you find the critical points of a function?
2. What is the significance of the first derivative test in determining increasing or decreasing intervals?
3. Can a function be increasing on some intervals and decreasing on others?
4. How does the degree of a polynomial affect the behavior of its graph?
5. How can you use the second derivative to determine concavity?

### Tip:
Always analyze the sign of the derivative to determine whether a function is increasing or decreasing.

Describe intervals for which the function is increasing, decreasing, or constant. 6. f(x) = 2x^2 7. f(x) = -3x^-5 8. g(x) = -x^3 ÷2

Determine the intervals where the functions f(x) = 2x^2, f(x) = -3x^-5, and g(x) = -x^3 ÷2 are increasing, decreasing, or constant. Learn how to use derivatives and critical points to analyze the behavior of these functions. Suitable for students in Grades 11-12.

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