AM ≥ GM Inequality

For any two positive numbers $a$ and $b$, the arithmetic mean is at least the geometric mean: $\frac{a+b}{2} \ge \sqrt{ab}$, with equality only when $a = b$. So for $a = 4$ and $b = 9$, $\mathrm{AM} = 6.5$ and $\mathrm{GM} = 6$, confirming $6.5 \ge 6$.

Slide positive values $a$ and $b$. The lab shows $\mathrm{AM}$, $\mathrm{GM}$, the gap $\mathrm{AM}-\mathrm{GM}\ge 0$, and harmonic mean $\mathrm{HM} = \frac{2ab}{a+b}$. Set $a = b$ to see all three means coincide — a classic equality case.

Maximizing $xy$ subject to $x+y = 10$: by AM-GM, $xy \le (x+y)^2/4 = 25$, attained when $x = y = 5$. JEE Main and Advanced lean heavily on this. Practising the lab builds the instinct to convert sums to products and vice versa.

Compounding returns: an investment doubling and halving over two years gives geometric mean return $\sqrt{2\times 0.5} = 1$, so net return is $0\%$ even though the arithmetic mean of $+100\%$ and $-50\%$ is $+25\%$. AM-GM warns investors against trusting simple averages of returns.