Suppose the function f is continuous on the interval [a, b] and never zero on [a, b].
Is it possible that f(z) < 0 for some z ∈ [a, b] and f(w) > 0 for some w ∈ [a, b]?
Explain your answer.
1
Expert's answer
2012-10-16T09:25:21-0400
This is not impossible due the intermediate value theorem. This theorem claims that if a continuous function f : [z,w] --> R on the interval [z,w] is such that f(z) and f(w) are non-zero and have distinct signs, e.g. either f(z)>0 and f(w)<0 or f(z)<0 and f(w)>0, then there exists a point c in [z,w] such that f(c)=0.
In our case suppose f(z)<0 for some z ∈ [a, b] and f(w)>0 for some w ∈ [a, b] We can assume that z<w. Then by the above theorem ther eshould exists a point c in [z,w] such that f(c)=0. But c also belongs to [a,b], and so f is not never zero on [a,b]. This gives a contradiction, and so
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