(i) If ` a + b + c = 5 , (a, b, c gt 0)`, find the greatest value of ` a^(3) b^(2) c^(4)` .
(ii) Find the greatest value of ` (5 - x)^(2) (x -2)^(2) ` , given ` 2 lt x lt 5 ` .

Explanation

To solve the given problems, we will use the method of inequalities, specifically the AM-GM (Arithmetic Mean-Geometric Mean) inequality.

### Part (i): Finding the greatest value of \( a^3 b^2 c^4 \) given \( a + b + c = 5 \) and \( a, b, c > 0 \)

1. **Express the terms**: We want to maximize \( a^3 b^2 c^4 \). We can rewrite \( a, b, c \) in a way that allows us to apply the AM-GM inequality:
\[ a^3 = a + a + a, \quad b^2 = b + b, \quad c^4 = c + c + c + c \]
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