Maximum & Minimum Values

At a glance 11 formulas

1. 1Vertex$ax^2 + bx + c:\ \text{extreme at } x = -\dfrac{b}{2a}$$a > 0:\ \text{least};\quad a < 0:\ \text{greatest}$
2. 2Extreme value$\text{value} = c - \dfrac{b^2}{4a}$$= \dfrac{4ac - b^2}{4a} \text{ for } ax^2 + bx + c$
3. 3Squares are never negative$(x - p)^2 + (y - q)^2 + k \ge k$$\text{least value } k \text{ at } x = p,\ y = q$
4. 4Whole numbers onlyIf $x$ must be a whole number, check the whole numbers on either side of $-\frac{b}{2a}$ and keep the better one.
5. 5Flip a reciprocal$f > 0:\ \text{greatest } \dfrac{k}{f} = \dfrac{k}{\text{least } f} \ (k > 0)$
6. 6Fixed sum$x + y = S \Rightarrow xy \le \dfrac{S^2}{4}$$\text{greatest when } x = y = \tfrac{S}{2}$
7. 7Fixed product$x + \dfrac{k}{x} \ge 2\sqrt{k} \ (x, k > 0)$$xy = P \Rightarrow x + y \ge 2\sqrt{P} \ (x, y > 0)$
8. 8Unequal powers$x + y = S:\ x^m y^n \text{ greatest at } \dfrac{x}{m} = \dfrac{y}{n}$$\dfrac{a + b + c}{3} \ge \sqrt[3]{abc} \ (a, b, c > 0)$
9. 9Fixed fence$\text{perimeter } P:\ \text{square, area } \dfrac{P^2}{16}$$\text{wall as one side, fence } L:\ \text{area } \dfrac{L^2}{8}$
10. 10Rectangle in a triangleThe largest rectangle with a corner at a right triangle's right angle takes half of each leg, and covers half the triangle's area.
11. 11A straight-line condition$ax + by = c \Rightarrow \text{least } x^2 + y^2 = \dfrac{c^2}{a^2 + b^2}$$\text{or substitute for one variable and use the vertex}$