have problems with this math. question?(Serious answers please).?

Question

Please give me a FORMAL proof. MANY THANKS.
Prove that if a≠0 then |a+(1/a)|≥2

Answer 1

Define function F on the non-zero reals by F(x)=|x+(1/x)|.
Because 1/a and a always have the same sign, for any positive number p, F(-p) = |-p+(1/(-p)) = |-[p+(1/p)]| = |p+(1/p)| = F(p).
The significance of this is that the minimum value of F(x) for negatiave values of x is the same as the minimum value of F(x) for positive values of x. So, it is enough to prove it for positive x. For positive values of x, F(x) = x+(1/x) = x+x^(-1) and the derivative F'(x) is 1-x^(-2) , where ^ means exponent.

F is continuous over the positive reals, so it has a relative extreme value when F'(x)=0. This has ONE positive solution, x=1, and for that value, F(x)=F(1)=1+(1/1)=2.
This extreme value for F is clearly a minimum, since any other specific positive x gives F(x) greater than 2. It is also an absolute minimum since F values for any chosen x on both sides of 1 are seen to be greater than 2.
Therefore F(x) never less than 2 for positive x, and so never less than 2 for non-zero x.
This completes the proof.

Answer 2

A formal proof makes this tough.

I would use some limits.

lets call your function A.

Lim(F(a))=infinity
(a approaches 0)

Lim(f(a))=infinity
(a approaches infinity)

You might be able to use l'hospital's on these, but i'm not inclined to, looks like it should just cancel out properly.

Logically, it's easy. The only way to make the fraction be less than one is to have A be larger than one, which in turn makes the equation true.

It is a catch 22. Tough with formalities, but easy with simple logic.

Answer 3

Multifply through by "a" and the put all terms on one side
This gives: a^2 - 2a + 1
Factor this to get: (a -1)^2

Since this is some number squared it is always greater than or equal to 0.