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[問題] HELP!!! Sequences, Functions and Derivatives

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HELP!!! Sequences, Functions and Derivatives

呢條唔係好識做
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This question shouldn't be too difficult if you could consider it from the perspective of analytic geometry. Since Q has to be a polynomial (with real coefficients) without real zeros, just take
it to be, say, x²+1. Now we have to ensure that for any x ∈ℝ P(x)/Q(x) ∈ [-1,1]. So in this case, we have to ensure that for any x ∈ℝ |P(x)| ≤ x²+1, or -x²−1≤ P(x) ≤ x²+1. Graphically, this means that the graph corresponding to the polynomial given by P(x) has to lie between the respective graphs corresponding to -x²−1 and x²+1. On the other hand, we have to ensure that there exist x′, x"∈ℝ such that P(x′)/Q(x′) = 1 and P(x")/Q(x") = -1. So the graph corresponding to the polynomial we need has to have intersections with both of the graphs corresponding to -x²−1 and x²+1. Thus a candidate for this polynomial would be 2x, which you can check on your own. Since both P and Q here are continuous on ℝ, this rational function will also be continuous on ℝ and assume any value in [-1,1]. Hope this can help.

[ 本帖最後由 markckw76 於 2012-11-26 05:51 AM 編輯 ]

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