The problem provides values of a continuous real function $f: \mathbb{R} \to \mathbb{R}$ at several points:

1. $f(-e) = -2.71$
2. $f(-0.1e) = 3.14$
3. $f(-0.01) = 0.05$
4. $f(0.01) = 27.18$
5. $f(1) = -0.027$
6. $f(e) = 5$
7. $f(3) = 5$
8. $f(2e) = -2$

The question asks for the minimum number of zeros of $f$ (values of $x$ for which $f(x) = 0$).

Solution Outline

To determine the minimum number of zeros, we can apply the Intermediate Value Theorem (IVT), which states that if $f$ is continuous on an interval $[a, b]$ and takes values of opposite signs at $a$ and $b$, then there exists at least one root $c \in (a, b)$ such that $f(c) = 0$.

Applying the Intermediate Value Theorem

Looking at the function values provided:

1. Between $f(-e) = -2.71$ and $f(-0.1e) = 3.14$, the function changes sign, so there is at least one root in this interval.
2. Between $f(1) = -0.027$ and $f(e) = 5$, the function again changes sign, indicating at least one more root in this interval.
3. Between $f(e) = 5$ and $f(2e) = -2$, the function changes sign once more, indicating at least one more root in this interval.

Conclusion

By applying the Intermediate Value Theorem in these intervals, we find at least three intervals where $f$ must cross the x-axis. Therefore, the function $f$ has at least 3 zeros.

Answer

The correct choice is:

• 3

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

Here are some related questions that expand on this topic:

1. What is the Intermediate Value Theorem, and how is it applied in different scenarios?
2. How can we determine if a function has exactly three zeros, rather than just at least three?
3. What are other methods to find the zeros of a function besides the Intermediate Value Theorem?
4. How does continuity of a function play a role in determining its roots?
5. Can the Intermediate Value Theorem be applied to non-continuous functions? Why or why not?

Tip: When analyzing function behavior across intervals, always check if the function changes sign. This change often hints at the existence of a root within the interval.