Next: 3.3.11 Concave Down Functions
Up: 3.3 Linear Interval Arithmetic
Previous: 3.3.9 Examples with a Piecewise
We will determine
for any concave up function .
Since g is concave up, .
Let ;
we assume that , so
we may take
.
A simple proof by contradiction, which follows, shows that
is an upper bound for g:
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 ,
,
and .
The assumptions made do not overly restrict the applicability of the proof.
-
If , consider g|[0,1] in place of g.
-
If , consider
in place of g,
such that g' is concave up.
If exists,
it may be taken for y; otherwise, a trivial upper bound may be used.
-
If , consider
in place of g,
such that g' is concave up.
If exists,
it may be taken for y; otherwise, a trivial upper bound may be used.
The bound is optimal, since may not be lowered.
Lowering would lower
or , invalidating
as an upper bound.
Next: 3.3.11 Concave Down Functions
Up: 3.3 Linear Interval Arithmetic
Previous: 3.3.9 Examples with a Piecewise