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Topic: re: inflection points
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Posts: 6
Registered: 12/6/04
re: inflection points
Posted: Oct 27, 1994 2:30 PM
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On Thu Oct 27, 1994 gjporter posted:

>The question is the same as the following:
>Let f(x)=x^2for x<=0 0 for 0<=x<=1 and (x-1)^2 for x>1
>Does the function have a relative minimum?

The function has a relative minimum at each point of the interval
[0, 1] but the first and second derivative tests fail.

A relative minimum of a function f occurs at a point
c iff f(c) <= f(x) for all x in some open interval
containing c.

However no critical point is situated such that f'(x) to the left
is negative while f'(x) to the right is positive nor is there any
critical point c so that f''(c) is positive. Since the
definition of a relative minimum is satisfied, f(c) IS a relative
minimum. Since the conditions of the theorems for the first and
second derivative tests are not met, they are inconclusive.

Contrasted with the original question on inflection points, the
definition of an inflection point is not met, i.e.,

A point (c, f(c)) is called a point of inflection for
the graph y = f(x) iff there is an open neighborhood
containing c over which f''(x) is positive on one side
of c and negative on the other side.

Since f''(x) = 0 on the segment connecting the two curved
portions of the graph, the definition doesn't hold and the curve
has no inflection points.

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