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IN SUMMARY
Key Idea
For a function f1x2, a critical number is a number, c, in the domain of f1x2
such that f ¿1x2 0 or f ¿ 1x2 is undefined. As a result 1c, f1c22 is called a critical
point and usually corresponds to local or absolute extrema.
Need to Know
First Derivative Test
Let c be a critical number of a function f.
When moving through x-values from left to right:
• if f ¿1x2 changes from negative to positive at c, then 1c, f 1c22 is a local
minimum of f.
• if f ¿1x2 changes from positive to negative at c, then 1c, f 1c22 is a local
maximum of f.
• if f ¿1x2 does not change its sign at c, then 1c, f 1c22 is neither a local minimum
or a local maximum.
Algorithm for Finding Local Maximum and Minimum Values of a Function f
1. Find critical numbers of the function (that is, determine where f ¿1x2 0 and
where f ¿1x2 is undefined) for all x-values in the domain of f.
2. Use the first derivative to analyze whether f is increasing or decreasing on
either side of each critical number.
3. Based upon your findings in step 2., conclude whether each critical number
locates a local maximum value of the function f, a local minimum value, or
neither.
Exercise 4.2
C
PART A
1. Explain what it means to determine the critical points of the graph of a
given function.
2. a. For the function y x 3 6x 2 , explain how you would find the critical
points.
b. Determine the critical points for , and then sketch the graph.
3. Find the critical points for each function. Use the first derivative test to determine
whether the critical point is a local maximum, local minimum, or neither.
a. b. f 1x2
2x c. y x 3 3x 2 1
x2 9
y x 4 8x 2 y x 3 6x 2
178 4.2 CRITICAL POINTS, LOCAL MAXIMA, AND LOCAL MINIMA
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