The Image Theorem:

The image theorem, a corollary of the intermediate value theorem, expresses the property that if f is continuous on the interval [a, b], then the image (the set of y-values) of f on [a,b] is all real numbers between the minimum of f(x) on [a,b], inclusive. USE THE EXTREME VALUE THEOREM AS A LEMMA (A PRELIMINARY RESULT) TO PROVE THE IMAGE THEOREM.

To prove the Image Theorem using the Extreme Value Theorem as a lemma, follow these steps:

1. Start by assuming that f is a continuous function on the interval [a, b].

2. Apply the Extreme Value Theorem to the function f on the interval [a, b]. The theorem states that since f is continuous on a closed and bounded interval, it must have a maximum and minimum value on that interval.

3. Let's denote the minimum of f(x) on the interval [a, b] as m. Hence, there exists at least one point, say c, in [a, b] where f(c) = m.

4. Now, we want to prove that the image of f on [a, b] is all real numbers between the minimum value, m, and some maximum value, M, which we will determine.

5. By applying the Extreme Value Theorem, we know that f has a maximum value on the interval [a, b]. Let's denote it as M.

6. To establish the Image Theorem, we need to show that for any given real number y between m and M, there exists some point x in [a, b] such that f(x) = y.

7. Consider any real number y such that m ≤ y ≤ M. Since y falls between the minimum and maximum values of f(x) on [a, b], we can conclude that there exist at least two points in [a, b], say x1 and x2, such that f(x1) ≤ y ≤ f(x2).

8. Now, with the Intermediate Value Theorem, we can state that since f is continuous on [a, b], there must exist some point x in [a, b] between x1 and x2 where f(x) = y.

9. Therefore, we have shown that for any real number y between the minimum value m and maximum value M, there exists at least one point x in [a, b] such that f(x) = y.

10. Consequently, the image (the set of y-values) of f on the interval [a, b] must contain all real numbers between the minimum value m and the maximum value M.

By using the lemma provided by the Extreme Value Theorem and applying the Intermediate Value Theorem, we have proved the Image Theorem, which states that if f is continuous on the interval [a, b], then the image of f on [a, b] is all real numbers between the minimum of f(x) on [a, b], inclusive.