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.