Final answer:
To prove the expression for any positive real numbers a and b, a + b is greater than 2√(ab), we apply the arithmetic mean-geometric mean inequality. We conclude that the inequality is strict unless a and b are equal.
Step-by-step explanation:
To prove that for any positive real numbers a and b, a + b is greater than 2√(ab), we can use the following algebraic manipulation:
- Start with the assumption that a and b are positive real numbers.
- Apply the arithmetic mean-geometric mean inequality which states that for non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean. This can be represented as (a+b)/2 ≥ √(ab).
- Multiply both sides by 2 to remove the fraction: a + b ≥ 2√(ab).
- However, equality holds if and only if a = b. If a ≠ b, then the inequality is strict: a + b > 2√(ab).
Thus, we have proved that for any positive real numbers a and b, a + b is strictly greater than 2√(ab) unless a and b are equal.