Page 1 | © 2014 by Janice L. Epstein 5.3 Extrema. Inflection Points, and Graphing 5.3: Extrema, Inflection Points, and Graphing
A critical number of a function f is a number c in the domain of f such
that either f ¢(c) = 0 or f ¢(c) does not exist.
First Derivative Test
Suppose that c is a critical number of a continuous function f
 If f ¢ changes from positive to negative at c, then f has a local
maximum at c.
 If f ¢ changes from negative to positive at c, then f has a local
minimum at c.
 If f ¢ does not change sign at c, then f does not have a local
maximum or minimum at c.
Example: Find all local and global extrema for the functions below using
the 1st DT
(a) f ( x ) = x 4 - 6 x 2 + 4
(b) f ( x) = x x + 1
(c) f ( x ) = xe 2 x
Page 2 | © 2014 by Janice L. Epstein 5.3 Extrema. Inflection Points, and Graphing Second Derivative Test
Suppose f ¢¢ is continuous near c.

If f ¢ (c ) = 0 and f ¢¢ (c ) > 0 , then f has a local minimum at c

If f ¢ (c ) = 0 and f ¢¢ (c ) < 0 , then f has a local maximum at c
Example: Use the 2nd DT to find the local extrema for the functions
(b) f ( x) = x x + 1
(c) f ( x ) = xe 2 x
(a) f ( x ) = x 4 - 6 x 2 + 4
Page 3 | © 2014 by Janice L. Epstein 5.3 Extrema. Inflection Points, and Graphing Checklist for Graphing a Function
A.Use f ( x ) to
1) Determine the domain of the function and the intervals on which the
function is continuous
2) Find all horizontal or oblique asymptotes
3) Find vertical asymptotes
4) Find x and y-intercepts
B. Use f ¢ ( x ) to
5) Find the critical values
6) Find intervals where the function is increasing and decreasing
7) Find all relative extrema
8) Find all global extrema with information from A and B6
C. Use f ¢¢ ( x) to
9) Find intervals where the function is concave up and concave down
10) Find all inflection points.
A line y  mx  b is an oblique asymptote if lim éë f ( x ) - (mx + b)ùû = 0
x¥
3x3  x 2  6 x  4
Example: Does f  x  
have an oblique asymptote? If
x2  2
yes, what is it?
Page 4 | © 2014 by Janice L. Epstein 5.3 Extrema. Inflection Points, and Graphing Example: Sketch f ( x ) = x 4 - 6 x 2 + 4 and f ( x) =
x
( x -1)
2