Bounds and Extrema

A competition problem may ask for the largest possible area or the smallest possible expression. You can often answer by proving a bound and showing exactly when it is reached.

What you will learn

• Derive useful inequalities
• Locate equality cases
• Optimize elementary expressions

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Start with something that cannot be negative

For every real $t$, its square is nonnegative. This elementary fact supplies many useful bounds. If you rewrite an expression as a constant plus a square, the constant is a lower bound. If you rewrite it as a constant minus a square, the constant is an upper bound. Equality occurs only when the square is zero.

Completing the square organizes a quadratic into this form. Take half the linear coefficient inside the square, then compensate for the constant you introduced. With a leading coefficient other than one, factor it from the quadratic and linear terms first. Expand your final expression as a quick check that the rewriting is exact.

Balanced quantities control products

For real $a,b$, the identity $(a-b)^2=(a+b)^2-4ab$ shows that $4ab\le(a+b)^2$. When the sum is fixed, this limits the product, and equality requires $a=b$. Geometrically, among rectangles with a fixed perimeter, the square has the greatest area. The conclusion follows from algebra rather than from how a sketch looks.

For positive quantities, the same reasoning gives the arithmetic-geometric mean inequality. Be attentive to the permitted domain: if lengths must be integers or one variable must lie in a certain interval, the balanced equality case may be unavailable. Then compare the nearest allowed candidates or analyze how far the expression lies from the ideal value.

Prove attainability, not just an estimate

A minimum is an allowed value that is no larger than any other allowed value. Merely proving an expression is at least some number does not show that number is the minimum; you must exhibit an input that attains it. This is particularly important for strict inequalities and domains that omit an endpoint.

Reciprocal expressions reward a similar structural approach. For $x>0$ and $k>0$, compare $x+k/x$ by observing that $(\sqrt{x}-\sqrt{k/x})^2\ge0$. This gives a lower bound and an equality condition simultaneously. In a contest solution, state the sign assumption, derive the bound, and check whether the equality input is permitted before announcing the result.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Announcing a bound without showing it can be attained.
• Applying a positive-variable inequality to a negative variable.
• Ignoring integer or interval restrictions on the equality case.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Rewrite a target using a nonnegative square.
• Identify the equality condition in the same calculation.
• Check that the equality input belongs to the allowed domain.