On Mon, 14 Jan 2008 08:06:04 -0800 (PST), Dougsd1r
<dougsdir@[EMAIL PROTECTED]
> wrote:
>My question is why create a full table of sign?
>
>here is another example where i have drawn of the table of sign
>
>http://i13.tinypic.com/80pwg74.gif
>
showing f'(x) = 4x(x-1)(x+1)
>it seems like all positives mean f'(x) is positives and all negatives
>mean f'(x) is negative but what about combinations of postives and
>negatives
>
It has long puzzled me why textbooks continue to suggest using "test
values" between the critical points to determine the sign of f'(x).
There is a much quicker way.
Since you have the factors, you have the stationary points 0, 1, and
-1 which are the only places where the corresponding factors may
change sign. So in the above example we notice that for x > 1 the
factors have signs + + +, and the sign of the product is determined by
the signs of the factors (positive)
As x decreases through 1 the middle factor changes sign:
+ - + (negative)
As x decreases through 0 the left factor changes:
- - + (positive)
As x decreases through -1 the right factor changes
- - - (negative)
If, for example the (x-1) factor were squared as: 4x(x-1)^2(x+1)
you might write the three cases as:
+ +^2 + (positive)
+ -^2 + (still positive)
- -^2 + (negative)
- -^2 - (positive)
You don't have to plug in any test values and the method works just as
well if there are factors in the denominator.
--Lynn
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