All books

Professional

Apps About Coach Log in Start reading

Mathematics · Glossary

What is Maximum, minimum?

Also known as: maximum · minimum

Definition 3.12 High School Mathematics · Chapter 3 — Functions

Let ff be defined on DD and a∈Da \in D. The value f(a)f(a) is the maximum of ff on DD when f(x)≤f(a)f(x) \leq f(a) for all x∈Dx \in D; it is the minimum when f(x)≥f(a)f(x) \geq f(a) for all x∈Dx \in D. On a graph, they are the highest and lowest points of the curve.

The curve of f(x) = (x-1)2 + 2 never goes below the dashed line y = 2, and touches it at x = 1: the minimum of f is 2.
The curve of f(x)=(x−1)2+2f(x) = (x-1)^2 + 2 never goes below the dashed line y=2y = 2, and touches it at x=1x = 1: the minimum of ff is 22.

Examples

Example 3.13

Show that f(x)=(x−1)2+2f(x) = (x - 1)^2 + 2 has minimum 22 on R\R, attained at x=1x = 1. For every xx, the square (x−1)2(x-1)^2 is ≥0\geq 0, so f(x)=(x−1)2+2≥2f(x) = (x-1)^2 + 2 \geq 2; and f(1)=0+2=2f(1) = 0 + 2 = 2, so the value 22 is actually reached. Both facts together prove that 22 is the minimum.

Example 3.10

Consider the function graphed below on [−2, 4]\intcc{-2}{4}.

A function defined on [-2, 4], increasing then decreasing.
A function defined on [−2, 4]\intcc{-2}{4}, increasing then decreasing.

Reading the graph: ff increases from f(−2)=−1.5f(-2) = -1.5 up to its maximum f(1)=3f(1) = 3, then decreases down to f(4)=−1.5f(4) = -1.5. Its variation table is

xx−2-21144
ff−1.5-1.5↗\nearrow33↘\searrow−1.5-1.5
Read in context →

Related terms