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OCW Scholar - Single Variable Calculus 14 Online
OpenStudy (anonymous):

Hey ppl :) Any hints or clues on how to solve the following problem? Suppose that f and g are two functions both continuous on the interval [a, b], and such that f(a) = g(b) = p and f(b) = g(a) = q where p does not equal to q. Sketch typical graphs of two such functions . Then apply the intermediate value theorem to the function h(x) = f(x) - g(x) to show that f(c) = g(c) at some point c of (a, b).

OpenStudy (jamesj):

Sure. Look at the function h = f - g. What is the sign of h at a and b; namely what is h(a) and h(b) with respect to zero: less than, more than, equal? It's not hard to see that if p > q then h(a) > 0 > h(b) and if q > p then the opposite is true. That should give you the set you need to now apply the IVT to the function h on the interval [a,b]

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