Proving inequalities

Japanese school year: Math II

What you learn

You will learn fundamental methods to prove inequalities, such as taking differences, completing squares, and applying the AM-GM inequality. These techniques are vital for finding bounds of functions and establishing mathematical proofs. Prior proficiency in basic inequality properties and algebraic manipulation is recommended.

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Key points

Squaring any real number $a$ always results in $0$ or more. It is widely used in inequality proofs, and equals $0$ only when $a = 0$ .

a^2 \ge 0 \quad (a \in \mathbb{R})

For positive numbers $a$ and $b$ , the arithmetic mean $(a + b)/2$ is always greater than or equal to the geometric mean $\sqrt{ab}$ . It is useful for finding minimum sums when the product is constant.

\frac{a + b}{2} \ge \sqrt{ab} \quad (a > 0, b > 0)

The arithmetic and geometric means are equal if and only if the two numbers $a$ and $b$ are identical. This is used to find when maximum or minimum values occur.

\frac{a + b}{2} = \sqrt{ab} \iff a = b

The Cauchy-Schwarz inequality holds for any real numbers $a, b, x$ , and $y$ . It connects sum of squares with linear sums, with equality holding when $ay = bx$ .

(a^2 + b^2)(x^2 + y^2) \ge (ax + by)^2

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Practice 1Coming soon