<s>[INST] Find the range of $f(x) = 3x^4 - 16x^3 + 18x^2 + 5$ without applying differential calculus 
Find the range of $f(x) = 3x^4 - 16x^3 + 18x^2 + 5$ without applying differential calculus.

I tried to express $$f(x)=3x^4-16x^3+18x^2+5=A(ax^2+bx+c)^2+B(ax^2+bx+c)+C $$ which is a quadratic in $ax^2+bx+c$ which itself is quadratic in $x$. Comparing coefficients, we get
$$Aa^2=3 \tag{1}$$
$$2abA=-16$$
$$A(b^2+2ac)+aB=18$$
$$2bcA+bB=0$$
$$Ac^2+Bc+C=5$$
But I felt its very lengthy to solve these equations. Any hints?
 [/INST] Let us consider the minimum of $f(x)$ as $k$. Then $f(x)-k$ has at least one repeated root. That is
$$f(x)-k=3x^4-16x^3+18x^2+5- k=3(x-a)^2(x^2+bx+c)$$
Expand and equate coefficients, you get $k=5-3a^2 c$, and 
$$3(b-2a)=-16 \\ 3(c+a^2-2ab)=18 \\ a(ab-2c)=0$$
From third equation if $a=0$, then $k=5$. For $a \neq 0$ combinig three equations gives $a^2=4a-3$. Thus $a=1, 3$. 
$a=1$ leads to $c=-\frac{5}{3}$, and $k=5-3(-\frac{5}{3})=10$.
$a=3$ gives $c=1$, and $k=5-27=-22$.
Therefore minimum is $k=-22$.
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