Maximum & minimum

Japanese school year: Math I

What you learn

You will learn how to find maximum and minimum values of quadratic functions, considering restricted domains and their positions relative to the axis of symmetry. This is widely used in real-world optimization problems such as maximizing area or profit. Prior mastery of completing the square and sketching parabolas is essential.

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Key points

Without domain restrictions, a parabola opening upward ( $a > 0$ ) reaches its minimum $q$ at the vertex, while opening downward ( $a < 0$ ) reaches its maximum $q$ there.

a > 0 \implies \min = q \; (x = p), \quad a < 0 \implies \max = q \; (x = p)

When $x$ is restricted to $s \le x \le t$ , the maximum and minimum values occur either at the boundary endpoints $f(s), f(t)$ or at the vertex $f(p)$ if within range.

\max, \min \in \{f(s), f(t), f(p)\}

Comparing the parabola's axis of symmetry with the midpoint $(s + t)/2$ reveals which endpoint reaches the maximum (or minimum) value.

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Practice 1Coming soon