Question

If \( a_1, a_2, a_3 \) be any positive real numbers, then which of the following statement is true?

• A. \( 3a_1a_2a_3 \leq a_1^2 + a_2^2 + a_3^2 \)
• B. \( a_1^2 + a_2^2 + a_3^2 \geq 3a_1a_2a_3 \)
• C. \( a_1a_2a_3 \geq \frac{a_1^2 + a_2^2 + a_3^2}{3} \)
• D. \( a_1 + a_2 + a_3 \geq \frac{a_1^2 + a_2^2 + a_3^2}{3} \)

Hint:

The AM-GM inequality states that for positive real numbers, the arithmetic mean is greater than or equal to the geometric mean.

Answer 1

The Correct Option is B

Step 1: Applying the AM-GM inequality.
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we know that for positive real numbers \( a_1, a_2, a_3 \), the arithmetic mean is always greater than or equal to the geometric mean. This gives the relation \( a_1^2 + a_2^2 + a_3^2 \geq 3a_1a_2a_3 \).

Step 2: Conclusion.
The correct statement is option (2).