In article
<549c7b39-82d3-4a3d-967a-3747240ef45c@[EMAIL PROTECTED]
>,
Dougsd1r <dougsdir@[EMAIL PROTECTED]
> wrote:
> My question is why create a full table of sign?
>
> Here is a sample question
>
> f(x) = 5x^3 - 3x^5
>
> calculate the coordindates of the stationary points and the nature of
> each stationary point.
>
> here is the solution
>
> f'(x) = 15x^2 - 15x^4
> f'(x) = 15x^2(1 - x^2)
> f'(x) = 15x^2(1+ x)(1 - x)
>
> so the x coordinates of the stationary points are 0, -1 and 1
>
> the stationary points are therefore (-1,-2) , (0,0) and (1,2) so
> far so good
>
> now in the example the text book answer draws a table of sign to
> determine the nature of the S.P.s
>
> http://i14.tinypic.com/6jq79kp.gif
>
> is there a reason for drawing out the sign for each factor of 15x^2 -
> 15x^4
>
> ie 15x^2 , (1+ x), (1 - x)
>
> when you can just calculate the sign of f'(x) using 15x^2 - 15x^4
The reason for using the factors is because it is usually easier. You
already have f'(x) factored: f'(x) = 15x^2(1 + x)(1 - x). To test the
sign of f'(x) between, say, 0 and 1, pick an "easy" point between 0 and
1 and test the signs of the factors. If you pick 1/2, 15x^2 is
positive, 1 + x is positive and 1 - x is positive; "(+)*(+)*(+) = (+)"
so f'(x) is positive and f(x) is increasing.
It is certainly OK to check f'(x) instead of the factors but frequently
you can check the factors in your head while you need to do a little
work to calculate f'(x). In this case, since 15x^2 is always positive,
you can ignore it altogether.
Note that the second factor in the first table should be 1 + x
not x - 1; the values in that row are correct (for 1 + x).
[...]
--
Paul Sperry
Columbia, SC (USA)
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