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.

$4-3b=a\Rightarrow a\cdot b^3=(4-3b)b^3=-3b^4+4b^3$.
Sometimes it is more convenient to set $f(x)=-3x^4+4x^3$ and study the behaviour of this function.
$f'(x)=-12x^3 +12x^2=12x^2(1-x)$ which means that $f'(x)>0$ only when $x<1$.
This means at $x=1$ our function $f(x)$ obtains its greatest value, $f(1)=1$.
Hence $f(x)\leq 1$ which implies also that $a\cdot b^3\leq 1$
