Question

Question 4* (Similar to 18.1) Suppose f is a continuous function on a closed interval [a, b]. In class, we proved that f attains its maximum on that interval, i.e. there exists Imar E la, so that f(Imar) > f(x) for all r E (a,b]. We didn't prove that f attains its minimum on the interval, but I claimed that the proof is similar. In fact, you can use the fact that f attains its maximum on any closed interval to quickly prove that f attains its minimum on any closed interval. Here is how: (a Suppose that f is a continuous function on a closed interval (a, b). Prove that -f is also a continuous function on this interval. (You may use that for any c ER the function g(r) = cis continuous.) (b) Show that if- f attains its maximum at a point then f attains its minimum at that point. (c) Use the fact that any continuous function on a closed interval attains its maximum to conclude that f attains its minimum on the interval [a, b].

Answer 1

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