Question

According to the Extreme Value Theorem, which of the following functions are guaranteed to attain absolute maxiμm and miniμm values on the interval 0, 1? Justify your answer. (You do not need to find the absolute maxiμm/miniμm values.)

a) f(x)
b) g(x)
c) h(x)
d) k(x)

Answer 1

Without specific functions provided, the Extreme Value Theorem ensures functions that are continuous on the closed interval [0,1] will have an absolute maximum and minimum. This guarantee does not apply if the functions are discontinuous or the interval is not closed.

Step-by-step explanation:

According to the Extreme Value Theorem, a continuous function on a closed interval [a,b] is guaranteed to attain an absolute maximum and minimum on that interval.

The question does not provide the specific functions f(x), g(x), h(x), and k(x), so a general answer can only be provided based on the properties required by this theorem. For a function to meet the criteria of the Extreme Value Theorem, it must be both continuous on the closed interval and the interval itself must be finite and closed, meaning it includes its endpoints.

If the functions f(x), g(x), h(x), and k(x) are continuous on the interval [0,1], then they are guaranteed to attain absolute maximum and minimum values on this interval according to the theorem. However, if any of these functions have discontinuities, or if we were dealing with an open interval (not including either 0 or 1), this guarantee would not apply.