The image you provided is a sign chart for a function $f(x)$ and its derivative $f'(x)$. The table gives important information about the intervals where the function is increasing, decreasing, and its critical points. Here's a breakdown of the table:

1. $x$-values: The critical points or points of interest in this chart are at $x = -2$, $x = -1$, and $x = 0$.

2. $f'(x)$ (Derivative of $f(x)$):

   • For $x < -2$, $f'(x) > 0$ (positive), meaning $f(x)$ is increasing.
   • At $x = -2$, there is a sign change from positive to negative, so $f'(x)$ changes from positive to negative, indicating a local maximum at $x = -2$.
   • For $-2 < x < -1$, $f'(x) < 0$ (negative), meaning $f(x)$ is decreasing.
   • At $x = -1$, there is no sign change, meaning the derivative is negative (no local minimum or maximum).
   • For $-1 < x < 0$, $f'(x) < 0$ (still negative), so $f(x)$ continues to decrease.
   • At $x = 0$, the derivative changes from negative to positive, so $f(x)$ has a local minimum at $x = 0$.
   • For $x > 0$, $f'(x) > 0$ (positive), meaning $f(x)$ is increasing.

3. $f(x)$ (The function's behavior):

   • For $x < -2$, $f(x) \to -\infty$ as $x \to -\infty$, and $f(x)$ is increasing towards 2.
   • At $x = -2$, $f(x) = 2$, indicating a local maximum.
   • For $-2 < x < -1$, $f(x)$ decreases towards negative infinity.
   • For $x = -1$, $f(x) = -\infty$ at this discontinuity.
   • For $-1 < x < 0$, $f(x)$ increases again towards 2.
   • At $x = 0$, $f(x) = 2$, indicating a local minimum.
   • For $x > 0$, $f(x) \to +\infty$ as $x \to +\infty$.

Summary of important points:

• Local maximum at $x = -2$, where $f(x) = 2$.
• Local minimum at $x = 0$, where $f(x) = 2$.
• Discontinuity at $x = -1$, where $f(x) = -\infty$.

Would you like further details or clarification on any part of this analysis?

Related Questions:

1. How does the derivative's sign affect the shape of a graph?
2. What are the conditions for a local maximum or minimum in terms of the derivative?
3. How do discontinuities affect the behavior of a function?
4. What is the significance of the critical points in the analysis of a function?
5. Can you explain the second derivative test for concavity?

Tip:

When analyzing the behavior of a function, always pay attention to the sign of the derivative and where it changes to determine intervals of increase or decrease.