What theorem can help identify if a function has any zeros within a specific interval?

To determine whether a function f(x) has any zeros within a given interval, you can use the Intermediate Value Theorem (IVT). This theorem is a fundamental principle in calculus that states:

• If a function f is continuous on the closed interval
  [a, b], and if f(a) and f(b) have opposite signs, then there exists at least one c in the interval
  (a, b) such that f(c) = 0.

Here’s how you can apply the theorem step by step:

1. Ensure Continuity: First, confirm that the function f(x) is continuous over the interval
   [a, b]. Discontinuities can affect the validity of the IVT.
2. Evaluate Endpoints: Calculate the values of the function at the endpoints:
• f(a) and f(b).
3. Check Signs: Examine the signs of f(a) and f(b):
• If f(a) > 0 and f(b) < 0, or f(a) < 0 and f(b) > 0, then, by the IVT, there exists at least one root c in (a, b).
• If both signs are positive or both are negative, then f(x) does not have a zero in that interval.

In conclusion, the Intermediate Value Theorem is a powerful tool that provides a straightforward method to find zeros of continuous functions. Just remember that it requires the function to be continuous within your specified interval and that the values at the endpoints must straddle zero!