On Fri, 4 Oct 2013, wilson wrote: > On Thu, 03 Oct 2013 21:05:50 -0400, stony <email@example.com> wrote:
> > Prove that: > > > > ((1/(a^2+1)) + ((1/(b^2+1)) + ((1/(c^2+1)) <= 1/2 > > > > <= - less than or equal to 1/2
> One reason you might be having a problem is that there is no restriction on > the values of a, b, and c. If they are all zero, the inequality is false. > Now, assume a, b, and c are all larger than three. Then the inequality is > true. (Why?) Inbetween 0 and 3 the inequality becomes dependent upon the > relationships between a, b, and c.
If the prosition is true, then 1/a^2 + 1 < the expression <= 1/2 and 1/(a^2 + 1) < 1/2 iff 2 < a^2 + 1 iff 1 < a^2 iff 1 < |a|.
If a = b = c, then expression = 3/(a^2 + 1) and 3/(a^2 + 1) <= 1/2 iff 6 <= a^2 + 1 iff 5 <= a^2 iff sqr 5 <= |a|.
The propsition is false, if any one of a,b or c is between -1 and 1 and true if all three are at least sqr 5 or at most -sqr 5.