<s>[INST] Find the minimum of $a^4+2b^4+3c^4$ Suppose  that $a, b,$ and $c$
are positive real numbers satisfying
$a+b+c=3$. Find the minimum of $$a^4+2b^4+3c^4$$
We know $f(x)=x^4$ is convex on the positive reals so by Jensen's Inequality, we have $$a^4+b^4+c^4\ge \frac{1}{27}$$
Hence $$a^4+2b^4+3c^4\ge \frac{1}{27}+b^4+2c^4$$
What now?
 [/INST] In fact if you use Lagrange multipliers you get:
$\vec{\nabla}(a^4+2b^4+3c^4)=(4a^3,8b^3,12c^3)\ \ \propto\ \ \vec{\nabla}(a+b+c-3)=(1,1,1)$ which gives
$$a^3=2b^3=3c^3$$
same as Macavity $a^3:b^3:c^3=1:\frac 12:\frac 13$ ratio presentation.
Since $x\mapsto x^3\ $ bijective this is also $\ a=\sqrt[3]{2}\,b=\sqrt[3]{3}\,c\ $ and the sum becomes $$a+b+c=\Big(1+\frac 1{\sqrt[3]{2}}+\frac 1{\sqrt[3]{3}}\Big)\,a=3$$
which reveal the origin of this second LHS term you were asking about.
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