I am working on a howework question, trying to prove the following: 5 a + b &gt; 4 a b , ,where a and b are positive real numbers.I've tried multiplying expression by a b , squaring both sides of the equation so far. In both cases, after re-factoring, I could not conclude that inequality holds. Can someone point me in the right direction?

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I am working on a howework question, trying to prove the following:  5  a  +  b  &amp;gt;  4      a    b    , ,where a and b are positive real numbers.I&#039;ve tried multiplying expression by       a    b   , squaring both sides of the equation so far. In both cases, after re-factoring, I could not conclude that inequality holds. Can someone point me in the right direction?

percolarse2rzd · Accepted Answer

Step 1In general you have to be a bit careful about squaring inequalities, but here there’s no problem: since we’re assuming that a and b are positive,   5  a  +  b is positive, and therefore   5  a  +  b  &amp;gt;  4      a    b   if and only if   (  5  a  +  b      )    2    &amp;gt;  16  a  b , i.e., if and only if   25      a    2    −  6  a  b  +      b    2    &amp;gt;  0 . Now think of       b    2    −  6  a  b  +  25      a    2   as a quadratic in b and complete the square:       b    2    −  6  a  b  +  25      a    2    =  (  b  −  3  a      )    2    +  16      a    2   . Clearly   (  b  −  3  a      )    2    ≥  0 , and since a is positive,       a    2    &amp;gt;  0 , so   (  b  −  3      )    2    +  16      a    2    &amp;gt;  0 .Retracing our steps, we see that   (  5  a  +  b      )    2    &amp;gt;  16  a  b and hence that   5  a  +  b  =      (    5    a    +    b          )      2        &amp;gt;  4      a    b   .

enclinesbnnbk · Accepted Answer

Step 1For any real x and y you have  2  x  y  =      x    2    +      y    2    −  (  x  −  y      )    2  taking   x  =  2      a   and   y  =      b   you get  2  (  2      a        b    )  =  4  a  +  b  −  (  2      a    −      b        )    2    ≤  4  a  +  b  &amp;lt;  5  a  +  b

I am working on a howework question, trying to prove the following: 5 a + b >

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