# 4.3: Maximum and Minimum Values

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Finding the maximum and minimum values of a function can be very useful in applications. This is usually referred to as "optimization." Later on, we will examine the use of optimization in application problems. For now, we will use graphs to find the maximum and minimum values of the function.

In the example below, the maximum function value in the region shown is 100 . This occurs where $$x=2.5$$ In the graph below, the function shows a maximum value of 5 at $$x=-1$$ and $$a$$ minimum value of -27 at $$x=3$$ We can use the graphing calculator to find maximum and minimum values. In calculus the maximum and minimum values can be found algebraically.

The maximum and minimum values are also helpful in determining where the function is increasing or decreasing. In the previous example: The function is increasing between $$x=-2$$ and $$x=-1,$$ then again from $$x=3$$ to $$x=4 .$$ So we would say that the intervals in which the function is increasing are $$-2 \leq x<-1$$ and $$3<x \leq 4$$ or, using interval notation, [-2,-1)$$\cup(3,4]$$
The interval in which the function is decreasing is $$-1<x<3$$ or (-1,3).

Determine the domain and range for each function given. Then determine the maximum and minimum values for the function and the intervals in which the function is increasing or decreasing.    This page titled 4.3: Maximum and Minimum Values is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Richard W. Beveridge.