<s>[INST] Find the minimum and maximum value of $F=|a-2b|+|b-2c|+|c-2a|$ 
Let $a, b, c \in \Bbb R$ satisfy $a^2+b^2+c^2=21$. Find the minimum and maximum value of $$F=|a-2b|+|b-2c|+|c-2a|$$


I found $7\le F  \le \sqrt{399}$ but couldn't prove it. I was thinking of the following inequality:
$$|x_1+x_2+\cdots+x_n|\le|x_1|+|x_2|+\cdots+|x_n|\le \sqrt{n \left(x_1^2+x_2^2+\cdots+x_n^2\right)}$$
but they are not really efficient. Does anyone know how to solve this problem or know where it first appeared?
 [/INST] 1) For the maximum
It is easy to prove that
$$F = p a + qb + r c$$
for some $p, q, r$ (dependent on $a, b, c$) with
$p^2 + q^2 + r^2 \le 19$.
(See the remarks at the end.)
Using Cauchy-Bunyakovsky-Schwarz, we have
$$pa + qb + rc \le \sqrt{(p^2 + q^2 + r^2)(a^2 + b^2 + c^2) } \le \sqrt{19 \cdot 21} = \sqrt{399}.$$
Also, when $a = \frac{3}{19}\sqrt{399}, b = -\frac{1}{19}\sqrt{399}, 
c = - \frac{3}{19}\sqrt{399}$,
we have
$a^2 + b^2 + c^2 = 21$
and
$F = |a - 2b| + |b - 2c| + |c - 2a| = \sqrt{399}$.
Thus, the maximum of $F$ is $\sqrt{399}$.
Remarks:
We split into eight cases.
If $a - 2b \ge 0, b - 2c \ge 0, c - 2a \ge 0$, we have
$$F = a - 2b + b - 2c + c - 2a
= -a - b - c.$$
If $a - 2b \ge 0, b - 2c \ge 0, c - 2a < 0$, we have
$$F = a - 2b + b - 2c - c + 2a
= 3a - b - 3c.$$
If $a - 2b \ge 0, b - 2c < 0,
c - 2a \ge 0$, we have
$$F = a - 2b - b + 2c + c - 2a = -a - 3b + 3c.$$
Similarly, we deal with the remaining $5$ cases.

$\phantom{2}$
2) For the minimum
Denote $x = a - 2b, y = b - 2c, z = c - 2a$.
We have
\begin{align*}
 F^2 &= (|x| + |y| + |z|)^2\\
 &= x^2 + y^2 + z^2 
 + 2|xy| + 2|yz| + 2|zx|\\
 &\ge x^2 + y^2 + z^2 + 2|xy + yz + zx|\\
 &= 105 - 4(ab + bc + ca) + |6(ab + bc + ca) - 84|.
\end{align*}
where we have used
$x^2 + y^2 + z^2 = 5(a^2 + b^2 + c^2) - 4(ab + bc + ca)$,
and $xy + yz + zx = 3(ab + bc + ca) - 2(a^2 + b^2 + c^2)
$.
If $6(ab + bc + ca) - 84 \ge 0$,
we have
\begin{align*}
 F^2 &\ge 105 - 4(ab + bc + ca) + 6(ab + bc + ca) - 84\\
 & = 21 + 2(ab + bc + ca)\\
 &\ge 21 + 2 \cdot \frac{84}{6}\\
 & = 49.
\end{align*}
If $6(ab + bc + ca) - 84 < 0$,
we have
\begin{align*}
 F^2 &\ge 105 - 4(ab + bc + ca) - 6(ab + bc + ca) + 84\\ 
 &= 189 - 10(ab + bc + ca) \\
 &\ge 189 - 10\cdot \frac{84}{6} \\
 &= 49.
\end{align*}
Thus, we have $F \ge 7$.
Also, when $a = 4, b = 2, c = 1$, we have
$a^2 + b^2 + c^2 = 21$ and $F = 7$.
Thus, the minimum of $F$ is $7$.
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