<s>[INST] Finding the greatest value of $a^2b^3c^2$ if $a+b+c=3$ and all numbers are positive Find the greatest value of $a^2b^3c^2$ if $a+b+c=3$ and all numbers are positive.
Here is my attempt using $\text{AM-GM inequality}$:
$$AM=\frac{a+b+c+a+b+c+b}{7}$$
$$GM=\sqrt[7]{a^2b^3c^2}$$
We have to find the maximum value of the expression under the radical of the $GM$.
$GM$ will be maximum if all terms are equal. Hence,
$$a=b=c=1$$
So the maximum value of the expression should be $1$.
However, this is wrong (at least according to the problem book where I found this question).
I can't figure out what I did wrong. Can you help me?
Source: Resonance DLPD Algebra for JEE Mains and Advanced. Exercise 1, Part II, D-4.
Motivation: I am trying to practice mathematics problems for the JEE Mains and Advanced.
 [/INST] By AM-GM $$3=2\cdot\frac{a}{2}+3\cdot\frac{b}{3}+2\cdot\frac{c}{2}\geq7\sqrt[7]{\left(\frac{a}{2}\right)^2\left(\frac{b}{3}\right)^3\left(\frac{c}{2}\right)^2}.$$
The equality occurs for $\frac{a}{2}=\frac{b}{3}=\frac{c}{2}$ and $a+b+c=3.$
Can you end it now?
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