Re: Smoothest function passing through n points

In article <1117032203.893719.167040@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
XGR131 <XGR131@xxxxxxxxx> wrote:
>and for illustraion let points 0,0 1,1 2,4 be given , what is f if f
>passes through all the given points and also if g too passes through
>all the points 0,0 1,1, 2,4 then f'=<g'

If I understand you correctly, you are saying that you want
to have two differentiable functions f and g, defined on an
interval [a,b], such that:
f(a) = g(a) f(b) = g(b) f'(x) <= g'(x) for all x in (a,b)
where <= means less or equal.

Suppose f and g are not identical. Then integrating the
supposed inequality on both sides over (a,b) we get:

f(b) - f(a) < g(b) - g(a)

which may be rearranged as f(b)-g(b) < f(a)-g(a) which says
0 < 0, a contradiction. Therefore your set of requirements
on f and g cannot be met simultaneously.

--
Rouben Rostamian
.

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