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?

Using Holder's inequality:
$$\left(a^4+2b^4+3c^4\right)\cdot\left(1+\frac1{\sqrt[3]2}+\frac1{\sqrt[3]3}\right)^3 \geqslant (a+b+c)^4=3^4$$
Equality is possible when $a^3:b^3:c^3 =1:\frac12:\frac13$, so this gives the minimum.
