Is it harmonic inequality?

Algebra Level 2

For all values of x x ,
x 2 + 2 x 2 + 1 ≥ 2 \frac{x^2 +2}{\sqrt{x^2+1 }} \ge 2

Conditionally True False True

Ossama Ismail by Ossama Ismail , 63, Egypt

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Ossama Ismail
Dec 25, 2017

x 2 + 2 x 2 + 1 = x 2 + 1 x 2 + 1 + 1 x 2 + 1 = x 2 + 1 + 1 x 2 + 1 ≥ 2 \frac{x^2 +2}{\sqrt{x^2+1 }} = \frac{x^2 +1}{\sqrt{x^2+1 }} + \frac{1}{\sqrt{x^2+1 }} = {\sqrt{x^2+1 }} + \frac{1}{\sqrt{x^2+1 }} \ge 2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

For completeness, you should show that a minimum of 2 is attainable.

Pi Han Goh - 3 years, 5 months ago

Log in to reply

That's not necessary. The problem only claims that the expression is greater than or equal to 2, and that's what the solution shows. The problem does not ask for a minimum value, so it doesn't matter whether equality is possible or not.

Jon Haussmann - 3 years, 2 months ago

Log in to reply

Ah my bad. I'm too used to "You must find the values after solving this classical inequalities question".

Pi Han Goh - 3 years, 2 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...