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Topic: Help Needed in math
Replies: 25   Last Post: May 2, 2014 6:01 PM

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quasi

Posts: 10,398
Registered: 7/15/05
Re: Help Needed in math
Posted: May 2, 2014 12:49 AM
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William Elliot wrote:
>quasi wrote:
>>...
>>Nevertheless, konyberg's argument is correct.

>
>His answer is correct. I don't see his reasoning
>
>How does f(x) relate to the missing problem?


This is all very elementary.

Let g(x) = a*x^2 - 5*x - 3, for some fixed a in R.

The problem was to find the largest integer value of a such that

g(x) < 0 for all x in R

Note that g(0) = -3 < 0, regardless of the value of a.

Let f(x) = (5x + 3)/(x^2), x != 0.

For x != 0,

g(x) < 0

<=> a < f(x)

Hence

g(x) < 0 for all x in R

<=> a < f(x) for all x in R with x != 0

Such a value of a exists iff f is bounded below.

As konyberg showed, f has a global minimum value of -25/12,
hence

g(x) < 0 for all x in R

<=> a < -25/12

It follows that the largest integer value of a such that

g(x) < 0 for all x in R

is -3.

quasi



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