Problem 27 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If three positive numbers \(a, b\) and \(c\) are in A.P. such that \(\mathrm{abc}=8\), then the minimum possible value of \(b\) is : [OnlineApril9, 2017] (a) 2 (b) \(4^{\frac{1}{3}}\) (c) \(4^{\frac{2}{3}}\) (d) 4

Step-by-Step Solution

Answer

The minimum possible value of \(b\) is 2.

Step 1: Understand Arithmetic Progression (A.P.)

In an arithmetic progression (A.P.), the three numbers can be represented by \(a = b-d\), \(b\), and \(c = b+d\), where \(d\) is the common difference.

Step 2: Write expression for product abc

Using the A.P. property, the product \(abc\) can be expressed as \((b-d)b(b+d)\). Simplifying this, we have \(b(b^2-d^2) = 8\).

Step 3: Apply AM-GM Inequality

To find the minimum value of \(b\), apply the Arithmetic Mean - Geometric Mean (AM-GM) inequality:\[ \frac{b-d + b + b+d}{3} \geq \, \sqrt[3]{(b-d) \cdot b \cdot (b+d)} \]\[ b \geq \, \sqrt[3]{8} \]\[ b \geq 2 \]

Step 4: Simplify for Minimum b

From the equation \(b^3 - b \cdot d^2 = 8\), we see that \(b\) must be such that it satisfies the inequality found earlier. The minimum value that \(b\) can take and satisfy the product condition while also being positive is \(2\).

Step 5: Verify by Substitution

Substitute \(b = 2\) into the product condition equation: \(abc = (2-d) \cdot 2 \cdot (2+d) = 8\).This simplifies to \(4 = 4\), confirming \(2^3 - 2d^2 = 8\) holds. Therefore, \(b = 2\) satisfies all conditions.

Key Concepts

AM-GM Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) inequality is a fundamental concept in mathematics. It states that for any non-negative real numbers, the arithmetic mean (average) is always greater than or equal to the geometric mean. This can be particularly useful in problems involving sequences or looking to find minimum or maximum values.

In the context of an arithmetic progression (A.P.), with the numbers expressed as \(a = b-d\), \(b\), and \(c = b+d\), the AM-GM inequality helps find the minimum value of \(b\). The inequality applied here is:

• Arithmetic mean of the numbers in A.P.: \(\frac{b-d + b + b+d}{3} = b\)
• Geometric mean: \(\sqrt[3]{(b-d) \cdot b \cdot (b+d)}\)

For the arithmetic mean to be greater or equal, the inequality: \[ b \geq \sqrt[3]{8} \] is derived, leading to \(b \geq 2\). This shows that the smallest possible value of \(b\) solving this inequality is \(2\). This insight is pivotal in solving minimum value problems where one has constraints on products, such as the one here.

Product of Sequences

In mathematics, the product of a sequence is found by multiplying all of its terms. When dealing with sequences like an arithmetic progression (A.P.), it's sometimes necessary to evaluate the product of these terms under given conditions.

For the sequence consisting of three terms: \(a = b-d\), \(b\), and \(c = b+d\), the product condition \(abc = 8\) can be expressed as:

• Expanded Product: \((b-d) \cdot b \cdot (b+d)\)
• Simplified Form: \(b(b^2-d^2) = 8\)

Understanding the product of sequences is crucial as it allows mathematicians to simplify complex expressions and solve for unknowns, as seen where \(b = 2\) satisfies \(b^3 - b \cdot d^2 = 8\). This simplification provides a pathway to solving the exercise, confirming that the values chosen must match the product condition.

Minimum Value Problem

Minimum value problems in mathematics aim to find the smallest possible value that a variable can take under given conditions or constraints. It usually involves understanding relationships and applying inequalities or calculus methods.

In this exercise, we have constraints in the form of both the arithmetic progression properties and the product \(abc = 8\). The goal is to find the minimum possible value of \(b\). Using AM-GM inequality simplifies the process, providing a lower bound for \(b\): \(b \geq 2\).

Key Steps in solving the Minimum Value Problem include:

• Define the relationship: Using known properties of the sequence, express \(abc\) in terms of \(b\) and \(d\).
• Apply inequalities: Utilizing AM-GM provided the minimum threshold for \(b\).
• Verify solution: Substitute \(b = 2\) back into the equation to ensure all conditions are satisfied.

These steps establish that \(b = 2\) is the minimum value satisfying all given problems in this specific sequence, confirming the application of these mathematical tools in practical problems.