Providing a proof or counter example.

[Dougggggg]

Homework Statement

Prove or give a counterexample: for all x > 0 we have x²+1< (x+1)²$$\leq$$ 2(x²+1)

Homework Equations

The Attempt at a Solution

I used my calculator to do the graph of all 3 functions and saw that the statement was always true (at least for x>0). So I couldn't see another proof method that would work so I just went ahead with the direct proof.

Kept rewriting the functions and narrowing down until I got to this below.

-(1/x) < 2 - (1/x) $$\leq$$ x

My intuition says that if x>0 that when x gets smaller and smaller, the first two terms get more and more negative. As x gets bigger and bigger, the term on the right becomes larger while the other only get slightly larger. However, I didn't see that as a proper "proof."

If it is any help, my class has learned these proof methods so far: Direct Proof, Counterexample, Contrapositive, Contradiction, and proof by cases.

Edit: Found something, I managed to do some algebra and rearranged it to the form of -2x<0<(x-1)(x-1) and pretend that second inequality includes "or equal to." There is a square on the right which will always be positive or 0 and in the problem it says x>0 so that first term will always be less than 0. Sweet.

If something is wrong in my logic please tell me.

 

[SammyS]

For what values of x are any pair of the functions equal?

 

[Dougggggg]

For what values of x are any pair of the functions equal?

Try x=1

 

[mistermath]

Did you do your algebra wrong?
First one is: x^2+1
Second one is: x^2 + 2x + 1
Third term is: 2x^2 + 2

Subtract x^2 from every one you get:

1 less than 2x+1 less than or equal to x^2+2

Subtract 1 from every one you get:

0 less than 2x less than or equal to x^2 + 1

Now I would prove all of them separately:
0 less than 2x
then
0 less than x^2 + 1
then
2x less than or equal to x^2 + 1

OR

0 < 2x
2x less than or equal to x^2+1

finally use transitive property to show 0

 

[Dougggggg]

From where you are, minus 2x. You have (-2x), then 0, then x^2 -2x + 1. Which factors into (x-1)(x-1) or (x-1)^2

 

[mistermath]

From where you are, minus 2x. You have (-2x), then 0, then x^2 -2x + 1. Which factors into (x-1)(x-1) or (x-1)^2

I like that. It makes it even easier because any number squared is greater than or equal to zero. And -2x is always less than zero for x > 0. Both should be pretty easy to prove.

 

[Dougggggg]

Yea, that is what I ended up doing, it was turned in yesterday at like 9 am.