# Simple proof help?

halo31
simple proof help??

## Homework Statement

The question states: Let rεQ+ Prove that if (r^2+1)/(r)≤1, then (r^2+2)/(r)≤2.
I wanted to prove it trivially by proving it is true for all Q(x). would this be a correct way?

## The Attempt at a Solution

Since (r^2+2)/(r)≤2= (r-1)^2+1≤ 0 it follows (r-1)^2≤-1 for all rεQ+. Therefore (r^2+2)/(r)≤2.

Homework Helper
Dearly Missed

The minimum of the function f(r) = (r^2 + 1)/r = r + (1/r) is f_min = 2, at r = 1. So, it is impossible to have f(r) <= 1 with r > 0.

RGV

halo31

I realized my mistake you can't prove it trivially. The way I proved it is wrong since (r-1)^2+1≥0 and not ≤0.

halo31

why am i getting different graphs when I graph r-(1\r)≤1 and r^2-r+1? I just multiplied through by r and moved everything around but still I am getting different equations.

Homework Helper

why am i getting different graphs when I graph r-(1\r)≤1 and r^2-r+1? I just multiplied through by r and moved everything around but still I am getting different equations.
What do you mean by "r-(1\r)≤1"? there is no "≤" in your second formula. If you dif mean r- (1/r)≤ 1, multiplying through by (positive) r gives r^2- 1≤ r and then "moving everything around" gives r^2- r- 1≤ 0, not r^2- r+ 1≤ 0.

halo31

for the r- (1/r)≤ 1 i get a hyperbola looking graph and r^2- r- 1≤ 0 I get a parobla with both being shaded.