Re: wims ineq
- From: Dave nothing_else <davenadamas@xxxxxxxxx>
- Date: Tue, 6 Oct 2009 11:52:34 -0700 (PDT)
On Oct 6, 12:22 pm, superpollo <u...@xxxxxxxxxxx> wrote:
Dave nothing_else wrote:
What is a "step", really? How many are involved in
noting that (-10x+3)/(x^2+4) =
3 - (1/17) - ( (10x+17)^2+9 )/( 34 (x^2+4) )
a step is an action as described in the drop down menu. can you give the
steps to get to your expression?
Yes, I saw the pull-down menu on the web page you mentioned.
It didn't allow me to rewrite your function in the way I
have done it (i.e. as 3 - (a sum of squares) ). I do like
puzzles, where the object is to accomplish something with a
certain limited set of tools, but this one was too vague for me.
can you give the steps to get to your expression?
If you mean, "how did you get that presentation of the function?"
the answer is that I did not use a particular algorithm. I used
some theorems of calculus to find the maximum of f, and then
when I knew g(x) = 3 - f(x) was everywhere positive, I used another
theorem to know it would be expressible as a sum of squares.
(Better, since g(x) was a rational function whose denominator
was already a sum of squares, I knew I just needed to write the
numerator that way. That numerator was a quadratic polynomial
whose roots were necessarily non-real, and (x-a-bi)(x-a+bi)=
(x-a)^2+b^2 is then a sum of squares.) These "steps" allow you
to write your function as f(x) = 3 - ( (3x+5)^2 + 2)/(3(x^2+4))
which is clearly sufficient for your purposes.
The variation I got came from looking for a positive number r
for which the zeros of 3 - r - f(x) were not only non-real but
also had rational real and imaginary parts. Sometimes it's hard
not to get carried away :-)
dave
.
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