quasi
Posts:
10,226
Registered:
7/15/05


Re: Help Needed in math
Posted:
May 2, 2014 12:49 AM


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

