If a1, a2, a3 .... an are positive real numbersIf a + b + c = 18 then maximum value of a3b2c will be (a, b, c ∈ +)

The problem is to maximize the expression a^3 * b^2 * c given that a + b + c = 18, where a, b, and c are positive real numbers.

This is a problem of optimization under constraints, and it can be solved using the method of Lagrange multipliers, but there is a simpler way to solve it using the AM-GM inequality (Arithmetic Mean - Geometric Mean inequality).

The AM-GM inequality states that for any positive real numbers x1, x2, ..., xn, the arithmetic mean is always greater than or equal to the geometric mean. That is:

(x1 + x2 + ... + xn) / n >= (x1 * x2 * ... * xn)^(1/n)

Equality holds if and only if all the numbers are equal.

In this problem, we can rewrite the expression a^3 * b^2 * c as a * a * a * b * b * c. We have 7 terms here, and their sum is 18 (since a + a + a + b + b + c = 18).

Applying the AM-GM inequality, we get:

(18 / 7) >= (a * a * a * b * b * c)^(1/7)

Raising both sides to the power of 7, we get:

(18 / 7)^7 >= a^3 * b^2 * c

So the maximum value of a^3 * b^2 * c is (18 / 7)^7.

Equality holds when a = a = a = b = b = c, which means a = b = c = 18 / 7.