For each value for a and b, being real numbers, a 2 + b 2 ≥ a bShould I solve this by replacing all possible real number formulas in these to be able to prove this? As an example for odd numbers, by replacing 2 x + 1, etc...?

Name: For each value for a and b, being real numbers, a 2 + b 2 ≥ a bShould I solve this by replacing all possible real number formulas in these to be able to prove this? As an example for odd numbers, by replacing 2 x + 1, etc...?
Uploaded: 2022-02-23T17:22:57+00:00
Description: For each value for a and b, being real numbers, a 2 + b 2 ≥ a bShould I solve this by replacing all possible real number formulas in these to be able to prove this? As an example for odd numbers, by replacing 2 x + 1, etc...?

Question

For each value for a and b, being real numbers,       a    2    +      b    2    ≥  a  bShould I solve this by replacing all possible real number formulas in these to be able to prove this? As an example for odd numbers, by replacing   2  x  +  1, etc...?

Calvin Maddox · Accepted Answer

Step 1Try to write it as       a    2    +      b    2    −  a  b  =            2              a        2            +      2              b        2            −      2      a      b        2    =                    a        2            +              b        2            +      (      a      −      b              )        2              2  .Step 2Now all three terms are   ≥  0, so you get       a    2    +      b    2    ≥  a  b.

Kareem Mejia · Accepted Answer

Step 1Alternative:Either   (  a  ×  b  ) is negative, or it isn&#039;t.            Case 1:             (      a      ×      b      )             is negative        _  Then,       a    2    +      b    2    ≥  0  &amp;gt;  (  a  ×  b  )  .Step 2            Case 2:             (      a      ×      b      )             is not negative        _  Then   2  (  a  ×  b  )  ≥  (  a  ×  b  )  .Further, since   (  a  −  b      )    2    ≥  0 you have that      a    2    +      b    2    ≥  2  (  a  ×  b  )  ≥  (  a  ×  b  )  .