<s>[INST] What is the minimum value of a radical sum? How would you find the minimum value of $\sqrt {x^2+16} + \sqrt {x^2-12x+37}$ through algebraic manipulation? Graphing will clearly show that the minimum value is 4.8, but how do you get the answer manually?
In general, how would you find the minimum value(if it exists) of the expression $\sqrt[n_1] {P_1(x)} + \sqrt[n_2] {P_2(x)} + \sqrt[n_3] {P_3(x)} + \cdots + \sqrt[n_i] {P_i(x)}$, where $P_1(x), P_2(x), P_3(x), \ldots, P_i(x)$ are polynomials, and $n_1, n_2, n_3, \ldots,n_i$ are whole numbers? On the other hand, how would you prove that no minimum value for such a sum exists?
 [/INST] Another way.
By C-S $$\sqrt{x^2+16}+\sqrt{x^2-12x+37}=$$
$$=\frac{1}{\sqrt{2.44}}\sqrt{(1.2^2+1)^2(x^2+4^2)}+\frac{1}{\sqrt{2.44}}\sqrt{(1.2^2+1^2)((6-x)^2+1^2)}\geq$$
$$=\frac{1}{\sqrt{2.44}}\left(1.2x+4+1.2(6-x)+1\right)=\frac{12.2}{\sqrt{2.44}}=\sqrt{61}.$$
The equality occurs for $(x,4)||(6-x,1)$ or for $x=4.8,$ which says that we got a minimal value.
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