The image shows a question related to determining the zeros of continuous functions $f(x)$ and $g(x)$ on the interval $[-1, 1]$. It provides a table with the values of $f(x)$ and $g(x)$ at specific points ($x = -1$, $x = 0$, and $x = 1$) and asks for the best description of the zeros of these functions.

Here’s a breakdown of the problem:

Table:

• At $x = -1$:
  • $f(x) = 3$
  • $g(x) = 1$
• At $x = 0$:
  • $f(x) = 0$
  • $g(x) = -1$
• At $x = 1$:
  • $f(x) = -2$
  • $g(x) = 1$

The options describe the behavior of zeros for the functions $f(x)$ and $g(x)$ on the interval $[-1, 1]$.

Options:

1. $f(x)$ will have a zero on $(-1, 1)$.
2. $f(x)$ will not have a zero on $(-1, 1)$.
3. $f(x)$ may or may not have a zero on $(-1, 1)$.
4. $g(x)$ will have a zero on $(-1, 1)$.
5. $g(x)$ will not have a zero on $(-1, 1)$.
6. $g(x)$ may or may not have a zero on $(-1, 1)$.

Analysis:

• $f(x)$ clearly has a zero at $x = 0$, since $f(0) = 0$. Also, because $f(-1) = 3$ and $f(1) = -2$, by the Intermediate Value Theorem (IVT), there must be at least one zero between $-1$ and $1$.

  • This eliminates options 2 and 3, confirming that option 1 is correct.

• For $g(x)$, it does not have a zero at the given points. The signs of $g(x)$ do not change between $-1$ and $1$, since both $g(-1) = 1$ and $g(1) = 1$ are positive. Therefore, by the IVT, $g(x)$ does not have a zero on the interval.

  • This confirms that option 5 is correct, while options 4 and 6 are eliminated.

Answer:

Based on this analysis, the correct answer is B: II and V.

Would you like further details or have any questions?

Here are five related questions to explore:

1. How does the Intermediate Value Theorem guarantee the existence of a zero?
2. Can a function have more than one zero on a given interval?
3. What happens if a function's values at the interval endpoints have the same sign?
4. How do we apply the IVT in practical situations beyond polynomial functions?
5. How does continuity affect the behavior of function zeros?

Tip: Always check both function values and continuity when using the IVT to locate zeros.