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
$a^2 + b^3 + c^4.$
Note that for the surface $a^2 + b^3 + c^4=k$ to have the minimum value $k$, it is tangential to the surface $a + b^2 + c^3 = \frac{325}{9}$. Calculate their normal vectors 
$(1,2b,3c^2)$ and $(2a,3b^2,4c^3)$, and match them to get

$$\frac{2a}1=\frac{3b}2=\frac{4c}3$$

Then, plug $b= \frac43a$ and $c=\frac32a$ into $a + b^2 + c^3 = \frac{325}{9}$ to obtain the tangential point at 

$$a=2,\&gt;\&gt;\&gt;b = \frac83,\&gt;\&gt;\&gt;c=3$$

As a result, the minimum value is

$$a^2 + b^3 + c^4=2^2+\left(\frac83\right)^3+3^4=\frac{2807}{27}$$
Have you tried considering the Lagrangean $$\mathcal L(a, b, c) = a^2 + b^3 + c^4 + \lambda\left(a + b^2 + c^3 - \frac{325}9\right)?$$