Calculus: Your homework this weekend is to head to the youtube link here and watch the video. The
notes for the video are below- please print them out and complete them along with the video. I will be
checking this on Monday, and we will continue onto a new topic at that time. THIS IS EXTREMEMLY
IMPORTANT. Please do not help your friends “cheat” by allowing them to copy your notes- in the end,
this will only hurt them as they will miss out on the information. The video is fairly short, please
participate.
Name: ____________________
Date: _________
Calculus Enriched
Section 4.1: Local Minima and Maxima
As we know, the derivative is a measure of the rate of change of a function over a particular
interval or at a particular point.
We can use this concept to find what are known as local minima or local maxima.
Consider the following graph for a pictorial representation. Two arrows point to a local
minimum and a local maximum. Label them appropriately:
The points on a function when the derivative is equal to zero (or does not exist) are known as
critical points. It is at these points that a local minimum or maximum could exist.
Example 1: Use the given graph to identify the critical points. Determine if they are local
minima, maxima, or neither:
a)
b)
c)
From a drawing, the local minima and maxima are easy to find. We can use derivatives and
some fancy algebra, however, to find local minima and maxima of functions where a picture is
not provided. There are two methods to doing this. The first one is demonstrated below.
1
Example 2: Find the local minima and maxima of the function 𝑓(𝑥) = 3 𝑥 3 + 3𝑥 2 + 8𝑥 − 5 by
using the first derivative.
Step 1: Find the derivative of the function.
Step 2: Find the critical points of the function (these points are where the derivative is equal to
0)
Step 3: Set up the intervals that are defined by the critical points. Use those intervals to test the
sign of the derivative function, f’(x).
Interval
(−∞, ___)
(_____, ______)
(_____, ∞)
Sign of f’(x)
Step 4: Use these intervals to determine if the function has a local minimum, maximum or
neither at the critical points. List the points (x,y) and label them appropriately.
This process is called the First Derivative Test and is formalized here:
Example 3: Use the first derivative test to find the local maxima and minima of the following
functions:
a) 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 8
b) 𝑔(𝑥) = 𝑥 3 − 12𝑥 2 − 60𝑥
c) ℎ(𝑡) = 𝑒 𝑡 − 8𝑡