Tough multivariable inequality: Minimize $a^2 + b^3 + c^4$ given $a + b^2 + c^3 = \frac{325}{9}$ Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b^2 + c^3 = \frac{325}{9}.$ Find the minimum value of
$a^2 + b^3 + c^4.$

Have you tried considering the Lagrangean $$\mathcal L(a, b, c) = a^2 + b^3 + c^4 + \lambda\left(a + b^2 + c^3 - \frac{325}9\right)?$$
