3.3.11 Concave Down Functions

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3.3.11 Concave Down Functions

We will determine for any concave down function . Since g is concave down, . Let ; we assume that , so we may take . A simple proof by contradiction, which follows, shows that is an upper bound for g, excepting :

Assume that there is a point such that . Let , so . Furthermore, and imply that .

A quick review of the

chart reveals that this situation is impossible. There is no

,

such that

since

,

. We may take

to be infinitesimal for g differentiable at

; this corresponds to having

match, at

, both the value and the derivative of g. For discrete g we may take

and

to be neighbours. With such a choice of

,

is an upper bound for g since x may not be a member of

.

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

For differentiable g and constant , the bound is optimal when ; for other reasonable choices of , the optimal bound is similarly easy to determine. See section for details.

Next: 3.3.12 Lower Bounds Up: 3.3 Linear Interval Arithmetic Previous: 3.3.10 Concave Up Functions

Jeff Tupper

March 1996