<s>[INST] 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
$a^2 + b^3 + c^4.$
 [/INST] 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 
$$a=2,\>\>\>b = \frac83,\>\>\>c=3$$
As a result, the minimum value is
$$a^2 + b^3 + c^4=2^2+\left(\frac83\right)^3+3^4=\frac{2807}{27}$$
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