Prove inequalities using algebraic manipulation, the fact that squares are non-negative, and standard results such as the arithmetic mean-geometric mean inequality

What this dot point is asking

NESA wants you to prove inequalities, not merely verify them numerically. You must build arguments from accepted facts, the most important being that a real square is never negative, and you must know and apply the arithmetic mean-geometric mean (AM-GM) inequality. Crucially, you must structure the argument forwards from known truths to the required result, never assuming what you are trying to prove.

The non-negative square: the central tool

Every real number squared is greater than or equal to zero:

with equality if and only if $x = 0$. Almost every Extension 2 inequality proof reduces to spotting the right square. For instance, to prove $a^2 + b^2 \ge 2ab$, begin with the true statement

Expanding gives $a^2 - 2ab + b^2 \ge 0$, and adding $2ab$ to both sides yields $a^2 + b^2 \ge 2ab$. Equality holds exactly when $a = b$.

The arithmetic mean-geometric mean inequality

For non-negative reals $a$ and $b$, the arithmetic mean is at least the geometric mean:

Proof: since $a, b \ge 0$, both $\sqrt{a}$ and $\sqrt{b}$ are real. Start from the true statement

Expanding gives $a - 2\sqrt{a}\sqrt{b} + b \ge 0$, that is $a + b \ge 2\sqrt{ab}$. Dividing by $2$ gives the result. Equality holds when $\sqrt{a} = \sqrt{b}$, i.e. $a = b$.

A frequently used corollary is that for any positive real $x$,

obtained by applying AM-GM to $a = x$ and $b = \frac{1}{x}$, since $\sqrt{x \cdot \frac{1}{x}} = 1$.

The forwards structure of a proof

The single most important discipline is direction. A valid proof of $L \ge R$ starts from a statement already known to be true and arrives at $L \ge R$. It is invalid to write $L \ge R$, manipulate it, reach a true statement, and declare victory: that only shows the target implies something true, not that it is true.

When working from the target is the natural way to discover the proof, do the scratch work, then reverse the steps for the final write-up. Each reversed step must itself be a valid implication. If a step involves multiplying by a quantity, check its sign, because multiplying an inequality by a negative number reverses it.

Manipulating a given inequality

Often you are handed an inequality and asked to deduce another. Permitted operations include adding the same quantity to both sides, multiplying both sides by a positive quantity (preserving direction) or a negative quantity (reversing direction), and adding two inequalities of the same direction. You may not in general subtract inequalities or multiply two inequalities unless all terms are known to be positive.