Problem 33 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a+b+c=3\) and \(a>0, b>0, c>0\), then the greatest value of \(a^{2} b^{3} c^{2}\) is (A) \(\frac{3^{10} \cdot 2^{4}}{7^{7}}\) (B) \(\frac{3^{9} \cdot 2^{4}}{7^{7}}\) (C) \(\frac{3^{8} \cdot 2^{4}}{7^{7}}\) (D) None of these

Step-by-Step Solution

Answer

The greatest value is \(\frac{3^{10} \cdot 2^{4}}{7^{7}}\), option (A).

Step 1: Use the AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that the arithmetic mean of non-negative numbers is greater than or equal to the geometric mean of the same numbers. In this problem, we will apply it to three variables, but first, we need to express the target function in terms of a simple form. Let's consider the function as a single variable expression through substitution or balancing of provided conditions.

Step 2: Simplify the Expression

Given the conditions and looking at the function to maximize, consider using equal distribution of variables respecting their exponents in the function. We take that if the sum of variables is constant, splitting them equally can be beneficial, especially when powers in the product are similar.

Step 3: Substitute and Equalize

To simplify calculations, let us substitute such that our expressions become symmetrical. Suppose all variables assume equal value, i.e., let \(a = b = c = 1\). This makes the total sum \(a + b + c = 3\) satisfied and ensures symmetry in the values we substitute into the polynomial \(a^2b^3c^2\) to maximize it.

Step 4: Calculate the Polynomial

Substituting \(a = 1, b = 1, c = 1\) into \(a^2b^3c^2\) gives us \((1)^2(1)^3(1)^2 = 1^7 = 1\). However, we attempt other combinations near to symmetry such that the given constraints hold. Given inequality \(a^2b^3c^2\) with \(a+b+c=3\), a more equalified configuration should be sought for true max of real evaluated possibilities.

Step 5: Optimal Variable Configuration

Try weighted allocation corresponding to exponential power representation, given \(a^2b^3c^2\). Suppose the distribution resembles the power ratios \((2/7, 3/7, 2/7)\) of total sum. Set \(a = \frac{2}{7} \times 3\), \(b = \frac{3}{7} \times 3\), \(c = \frac{2}{7} \times 3\). Recalculate \(a, b, c\) as \(\frac{6}{7}, \frac{9}{7}, \frac{6}{7}\) respectively.

Step 6: Recalculate Given Optimal Values

Substitute the optimal \(a, b, c\) values into \(a^2 * b^3 * c^2\): \[\left(\frac{6}{7}\right)^2 \times \left(\frac{9}{7}\right)^3 \times \left(\frac{6}{7}\right)^2 = \frac{6^2 \cdot 9^3 \cdot 6^2}{7^7}\]. Simplifying gives the maximal value formula.

Step 7: Identify the Exact Numeric Answer

The calculated expression \[\frac{6^4 \cdot 9^3}{7^7} = \frac{3^{10} \cdot 2^4}{7^7}\], which identifies that the correct option is (A).

Key Concepts

Arithmetic Mean

The Arithmetic Mean (AM) is simply the average of a set of numbers. By adding all numbers in the set together and dividing by the number of values, we find the average. In mathematical terms, if we have values \(x_1, x_2, \ldots, x_n\), the arithmetic mean can be expressed as:
\[ \text{AM} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]

• This concept helps us understand how numbers balance out overall.
• It acts as a central value representing the dataset.
• In the context of maximization problems, applying AM can sometimes suggest an equal distribution of variables for optimization.

In the provided exercise, the sum \(a + b + c = 3\) represents a constant arithmetic total. With this in mind, the AM-GM inequality instructs us to leverage this equality to spread values equally for an optimal product.

Geometric Mean

The Geometric Mean (GM) is another way to determine the average of a set of numbers, typically used when the numbers are multiplicatively related. To calculate the geometric mean of \(n\) numbers, we take the nth root of their product:
\[ \text{GM} = \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n} \]

• It's particularly useful when working with exponential or growth rates.
• GM gives a more accurate mean for values that are scales or ratios rather than linear additions.
• It's a measure sensitive to balancing multiplicative differences among values.

In the exercise, we used GM by considering the function to maximize—\(a^2b^3c^2\). Applying GM implications, one way to think about maximizing it under the given constraint \(a+b+c=3\) is to try and achieve balance by equating their ratios, paving the way for simplification.

Inequality Theorems

Inequality theorems, like the AM-GM Inequality, offer methods to compare various means. The AM-GM Inequality states that for non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean:
\[ \frac{x_1 + x_2 + \ldots + x_n}{n} \geq \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n} \]

• The equality holds only when all \(x_i\) values are equal.
• The inequality helps in demonstrating how balanced or skewed distributions can affect average outcomes.
• In optimization problems, this theorem suggests the configuration for achieving maximum potential under equality constrains.

Applying the AM-GM Inequality to the problem, it suggested making the terms \(a, b, c\) as close to each other as possible. Once these values respect the weighted power ratios, further enhancements could be driven by assigning exact weight-distributed values, like \((2/7, 3/7, 2/7)\) for \(a, b, c\) respectively as found in the solution.