Question 1177684
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Tutor ikleyn's answer is 100% correct.  I wrote down a slightly different proof and thought I'd share, nothing earth-shattering.

Please note that the problem wording should be "...a and b are nonnegative real numbers..."   as "positive x" implies x>0.

Assume, WLOG, {{{a>=b}}}

Then a=b+d for some {{{d>=0}}} 

{{{a^4+b^4 = (b+d)^4 + b^4 }}}
= {{{ 2b^4+4b^3d+6b^2d^2+4bd^3+d^4 }}}   (1)

and
{{{ a^3b+ab^3 = (b+d)^3b + (b+d)b^3 = 2b^4+4b^3d+3b^2d^2+bd^3}}}   (2)

Now subtract (2) from (1) to get: 
{{{ (a^4+b^4)-(a^3b+ab^3) = 3b^2d^2+3bd^3+d^4  >= 0}}}