Suppose that f: [a,b] ---> R and g: [a,b]---> R are continuous functions such that f(a) ≤ g(a) and g(b) ≤ f(b). Prove that f(c) = g(c) for some c ∈ [a,b].
I would appreciate your help. Thanks in advance for your help and time.
I would appreciate your help. Thanks in advance for your help and time.
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Consider the function h:[a,b]->R given by h(x) = f(x) - g(x).
Since f and g are continuous functions, h must also be continuous which means the intermediate value theorem holds.
Now, if f(a) = g(a) or f(b) = g(b), then set c = a or c = b and we're done. Otherwise, we note that
h(a) < 0 and
h(b) > 0
by the intermediate value theorem, there must be a c in [a,b] such that h(c) = 0. If h(c) = 0, the f(c) = g(c).
In any case, there exists a c such that f(c) = g(c)
Since f and g are continuous functions, h must also be continuous which means the intermediate value theorem holds.
Now, if f(a) = g(a) or f(b) = g(b), then set c = a or c = b and we're done. Otherwise, we note that
h(a) < 0 and
h(b) > 0
by the intermediate value theorem, there must be a c in [a,b] such that h(c) = 0. If h(c) = 0, the f(c) = g(c).
In any case, there exists a c such that f(c) = g(c)