The image contains a multiple-choice question related to the Intermediate Value Theorem (IVT). The question asks which of the given functions must have a root within a specified interval, based on the IVT.

Here's a breakdown of the question and choices:

Question:
"Which of the following statements is guaranteed to be correct because of the Intermediate Value Theorem?"

Choices:

1. $f(x) = \frac{10x - 10}{x}$ has at least one root in the interval $(10, 12)$ for $x > 0$.
2. $f(x) = x^2 - 81$ has at least one root in the interval $(10, 12)$ for $x > 0$.
3. $f(x) = x^2 - 19$ has at least one root in the interval $(10, 12)$ for $x > 0$.
4. $f(x) = x^3 - 7$ has at least one root in the interval $(10, 12)$ for $x > 0$.
5. None of the above.

Solution:

The Intermediate Value Theorem states that if a function $f$ is continuous on the closed interval $[a, b]$ and takes different signs at $f(a)$ and $f(b)$, then there exists at least one $c \in (a, b)$ such that ( f(c) = 0 \