The given table provides values of a continuous function $f(x)$ at specific $x$-values, and we are asked to use the Intermediate Value Theorem (IVT) to determine the smallest interval where the function must have a root (i.e., where $f(x) = 0$).

Intermediate Value Theorem (IVT):

The IVT states that if a function $f(x)$ is continuous on a closed interval $[a, b]$ and if $f(a)$ and $f(b)$ have opposite signs (i.e., one is positive and the other is negative), then there exists at least one $c \in (a, b)$ such that $f(c) = 0$.

Table Data:

Solution:

We need to identify where the sign of $f(x)$ changes from negative to positive, as this indicates a root within that interval. Observing the table:

• $f(-3) = -3.05$ (negative) and $f(-2) = 3.01$ (positive).

This sign change between $x = -3$ and $x = -2$ indicates that the function must have a root in the interval $(-3, -2)$.

Conclusion:

The smallest interval in which the function must have a root is $\boxed{(-3, -2)}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the Intermediate Value Theorem apply to other types of functions?
2. Can a function have more than one root in the same interval?
3. How do we determine if there are multiple roots between two points?
4. What happens if the function is not continuous on the interval?
5. How can we approximate the root using numerical methods?

Tip: The IVT is a useful tool for identifying roots, but remember that it guarantees only the existence of at least one root in the interval, not the exact location.