The AM-GM INEQUALITY for two POSITIVE numbers $a$ and $b$ states that

1. Your syllabus: for any two POSITIVE numbers $a$ and $b$: 2. $\dfrac{a+b}{2} \geq \sqrt{ab}$ — i.e., AM $\geq$ GM. 3. EQUALITY holds if and only if $a = b$. 4. Proof sketch: $(a - b)^2 \geq 0$ (always, since square is non-negative) $\Rightarrow a^2 + b^2 \geq 2ab \Rightarrow a^2 + 2ab + b^2 \geq 4ab \Rightarrow (a+b)^2 \geq 4ab$ so $\sqrt{(a+b)^2/4} \geq \sqrt{ab}$, i.e. AM $\geq$ GM. 5. Example check: $a=2, b=8$: AM=5, GM=4. $5 \geq 4$ ✓ (strict inequality since $a \neq b$). 6. Option A is wrong unless $a=b$. Option B is the OPPOSITE inequality. Option D is much too strong. _Source: syllabus Mathematics, Ch 8, §8.4 (AM-GM relationship), p. 10–11._