Suppose f and g :[0,l)->R are continuous, concave and increasing where l<oo. Can we claim that they intersect at most finitely many points? What if we replace l with oo?

Kayla Mcdowell

Kayla Mcdowell

Answered question

2022-10-29

Suppose f and : [ 0 , l ) R are continuous, concave and increasing where l < . Can we claim that they intersect at most finitely many points? What if we replace l with ?

Answer & Explanation

Amadek6

Amadek6

Beginner2022-10-30Added 21 answers

No. Start with some continuous, concave and increasing f and find the points ( l 2 , f ( l 2 ) ) , ( l 4 , f ( l 4 ) ) , ( l 8 , f ( l 8 ) ) , ...
Then draw a continuous, concave and increasing function g through these points which is not identical everywhere to f; this will almost always be possible with some exceptions when f is only "weakly concave" and "weakly increasing".
For replacing l with , chose an arbitrary positive l and repeat.

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