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Next: 3.3.11 Concave Down Functions Up: 3.3 Linear Interval Arithmetic Previous: 3.3.9 Examples with a Piecewise

3.3.10 Concave Up Functions

We will determine tex2html_wrap_inline35701 for any concave up function tex2html_wrap_inline34165 . Since g is concave up, tex2html_wrap_inline36053 . Let tex2html_wrap_inline36055 ; we assume that tex2html_wrap_inline36057 , so we may take tex2html_wrap_inline36059 . A simple proof by contradiction, which follows, shows that tex2html_wrap_inline34669 is an upper bound for g:

math13195

Assume that there is a point tex2html_wrap_inline34333 such that tex2html_wrap_inline34675 . Let tex2html_wrap_inline34677 , so tex2html_wrap_inline35585 . Furthermore, tex2html_wrap_inline34251 and tex2html_wrap_inline36053 imply that tex2html_wrap_inline35629 .

figure18870

A quick review of the tex2html_wrap_inline35653 chart reveals that this situation is impossible. There is no tex2html_wrap_inline34333 such that tex2html_wrap_inline34675 since tex2html_wrap_inline34231 , tex2html_wrap_inline36059 , and tex2html_wrap_inline36103 .

The assumptions made do not overly restrict the applicability of the proof.

The bound is optimal, since tex2html_wrap_inline34669 may not be lowered. Lowering tex2html_wrap_inline34669 would lower tex2html_wrap_inline36139 or tex2html_wrap_inline36141 , invalidating tex2html_wrap_inline34669 as an upper bound.


next up previous notation contents
Next: 3.3.11 Concave Down Functions Up: 3.3 Linear Interval Arithmetic Previous: 3.3.9 Examples with a Piecewise
Jeff TupperMarch 1996