To determine which prediction is valid for the continuous function $f(x)$, based on the given data, we need to look at the values of $f(x)$ at specific points and analyze the sign of $f(x)$ over the intervals:

Here's a summary of the $f(x)$ values provided:

• $f(-3) = -15$
• $f(-2) = -5$
• $f(-1) = 0$
• $f(0) = 5$
• $f(1) = 0$
• $f(2) = -5$

Observations:

1. From $x = -1$ to $x = 0$, $f(x)$ is positive as it changes from 0 to 5.
2. From $x = 1$ to $x = 2$, $f(x)$ is negative as it changes from 0 to -5.

Let's examine each prediction:

• $f(x) \leq 0$ over the interval $(-\infty, \infty)$: This is false because $f(0) = 5$ (positive).
• $f(x) > 0$ over the interval $(-1, \infty)$: This is false because $f(x)$ takes negative values and zero values after $x = -1$.
• $f(x) \geq 0$ over the interval $[-1, 1]$: This is false because while $f(x)$ is non-negative at $x = -1, 0, 1$, it includes positive value at $x = 0$.
• $f(x) < 0$ over the interval $(0, 2)$: This is true. Both $f(1)$ and $f(2)$ are non-positive, with $f(1) = 0$ and $f(2) = -5$.

Thus, the valid prediction about the continuous function $f(x)$ is:

• $f(x) < 0$ over the interval $(0, 2)$.

Would you like more details on how these conclusions were reached, or do you have any other questions?

Here are five related questions you could explore:

1. What would be the behavior of $f(x)$ beyond the given points, such as $x > 2$ or $x < -3$?
2. Can you determine the possible type of function $f(x)$ could be based on the given values?
3. How does the sign of $f(x)$ change around the zero points at $x = -1$ and $x = 1$?
4. How would the graph of $f(x)$ look based on the values given?
5. What mathematical properties (like continuity or differentiability) can be inferred from the values of $f(x)$?

Tip: When trying to analyze a function's behavior over an interval, consider plotting the points and connecting them to visualize changes in the function's sign and value.