Volcanic Inequality!

Was not able to find an algebraic solution, so here's a calculus one.

Write the expression as f(a,b,c,d). f is smooth on the set S = {{a,b,c,d} | a+b+c+d = 4 and each is >=0}

Assume minimum occurs in interior of S. then we can find that point with lagrange multipliers. But with counterexamples we can show that those points are not minima. So the minima must occur on the boundary of S, WLOG when d=0.

We then want to minimize

$\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{4}$

with a+b+c=4.

We can perform Lagrange multipliers again to show the minimum does not occur in the interior of the new region defined by a+b+c=4. So WLOG we can assume c = 0 or b=0.

We then want to minimize

$\frac{a}{b^3+4}+\frac{b}{4}$

or

$\frac{a}{4}+\frac{c}{4}$

subject to a+b=4 or a+c=4.

The second one gives 1.

And by using standard calculus methods, the first one gives 2/3 when a=b=2. So 2/3 is our minimum.