Question

If \(a > 0, \, b > 0, \, c > 0\) and \(a, b, c\) are distinct, then \((a + b)(b + c)(c + a)\) is greater than

• A. \(2(a + b + c)\)
• B. \(3(a + b + c)\)
• C. \(6 abc\)
• D. \(8 abc\)

Hint:

In competitive exams, if you see products like \((a+b)(b+c)(c+a)\) with positive numbers, the AM-GM inequality is usually the first tool to use. If the numbers are not specified as distinct, the answer might be \(\geq 8abc\).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any set of non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. For distinct numbers, the AM is strictly greater than the GM.
Step 2: Key Formula or Approach:
For two positive numbers \( x \) and \( y \): \[ \frac{x+y}{2}>\sqrt{xy} \implies (x+y)>2\sqrt{xy} \]
Step 3: Detailed Explanation:
1. Apply the AM-GM inequality to each pair of terms individually: - For \( a \) and \( b \): \( a+b>2\sqrt{ab} \)
- For \( b \) and \( c \): \( b+c>2\sqrt{bc} \)
- For \( c \) and \( a \): \( c+a>2\sqrt{ca} \)
2. Multiply these three inequalities together: \[ (a+b)(b+c)(c+a)>(2\sqrt{ab})(2\sqrt{bc})(2\sqrt{ca}) \]
3. Simplify the right-hand side: \[ (a+b)(b+c)(c+a)>8\sqrt{a^2 b^2 c^2} \] \[ (a+b)(b+c)(c+a)>8abc \]
Step 4: Final Answer
The product is greater than 8abc.