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. \( 6abc \)
• D. \( 8abc \)

Hint:

If you forget the inequality, try small distinct positive values. Let \( a=1, b=2, c=3 \). Then \( (3)(5)(4) = 60 \). Option D: \( 8(1)(2)(3) = 48 \). Since \( 60>48 \), D is the likely candidate.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem involves the AM-GM Inequality (Arithmetic Mean - Geometric Mean Inequality), which states that for any set of non-negative real numbers, the AM is always greater than or equal to the GM. For distinct numbers, the AM is strictly greater than the GM.

Step 2: Key Formula or Approach:
For two positive distinct numbers \( x \) and \( y \): \[ \frac{x + y}{2}>\sqrt{xy} \implies (x + y)>2\sqrt{xy} \]

Step 3: Detailed Explanation:
Apply the AM-GM inequality to each pair of terms: 1. For \( a \) and \( b \): \( (a + b)>2\sqrt{ab} \) 2. For \( b \) and \( c \): \( (b + c)>2\sqrt{bc} \) 3. For \( c \) and \( a \): \( (c + a)>2\sqrt{ca} \) Multiplying these three inequalities together (since all terms are positive): \[ (a + b)(b + c)(c + a)>(2\sqrt{ab})(2\sqrt{bc})(2\sqrt{ca}) \] \[ (a + b)(b + c)(c + a)>8\sqrt{a^2 b^2 c^2} \] \[ (a + b)(b + c)(c + a)>8abc \]

Step 4: Final Answer
The expression is strictly greater than \( 8abc \).