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If a,b,c belong to N and 2a b c=12 then the maximum value of a^3bc^2 is equal to?
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If a,b,c belong to N and 2a b c=12 then the maximum value of a^3bc^2 i...
Given Information:

a, b, and c belong to the set of natural numbers (N).

2a + b + c = 12


Objective:

To find the maximum value of a^3bc^2.


Solution:

Let's start by rearranging the given equation:

2a = 12 - b - c


Step 1: Determine the Range of Values for a

In order to maximize the value of a^3bc^2, we need to determine the range of values for a.

The smallest value of a occurs when b and c are at their maximum possible values, which is 6 (since a, b, and c are natural numbers and their sum is 12). In this case, 2a = 12 - 6 - 6 = 0, so a = 0.

The largest value of a occurs when b and c are at their minimum values, which is 1. In this case, 2a = 12 - 1 - 1 = 10, so a = 5.

Therefore, the range of values for a is 0 ≤ a ≤ 5.


Step 2: Determine the Range of Values for b and c

Since a, b, and c are natural numbers, the range of values for b and c is 1 ≤ b, c ≤ 6.


Step 3: Evaluating a^3bc^2

Let's evaluate a^3bc^2 for each possible combination of values for a, b, and c within the given ranges:

a = 0, b = 6, c = 6: (0)^3 * 6 * (6)^2 = 0

a = 5, b = 1, c = 1: (5)^3 * 1 * (1)^2 = 125

For all other combinations of values, a is less than 5, so a^3bc^2 will be less than 125.


Step 4: Determining the Maximum Value

From the above evaluations, we can see that the maximum value of a^3bc^2 is 125, which occurs when a = 5, b = 1, and c = 1.


Answer:

The maximum value of a^3bc^2 is 125 when a = 5, b = 1, and c = 1.
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If a,b,c belong to N and 2a b c=12 then the maximum value of a^3bc^2 is equal to?
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