What is max($ab^3$) if $a+3b=4$? This question was  asked by my professor,i think in the set $\mathbb Z$,its answer is 1.But,he advised me for the further exploration in  $\mathbb  R$.But,what i think(i may be wrong) that in $\mathbb R
$ its answer is not unique,but it should be unique.
Any suggestions are heartly welcome.

If $a,b>0$, then $$a+3b=4 \ge 4\sqrt[3]{ab^3}$$From the mean inequality.  Thus $ab^3 \le 1$. 
If $ab<0$, then $ab^3<0$. Thus the maximum of $ab$ cannot be find here. 
If $a,b<0$, then $a+3b<4$. Thus a contradiction.  
