Math 165 – Quiz 7D,
increasing/decreasing/FDT – solutions
Problem 1 An unknown function f (x) has the derivative
f 0 (x) = (x + 1)(x − 2)(x − 4)2 (x − 6)(2 + cos x).
a) What are the critical points of f (x)?
b) Find the open intervals on which f (x) is increasing, and the open intervals
on which f (x) is decreasing. Indicate the intervals on a number line, labeled
with + and − signs.
c) At which points, if any, does f (x) have local maxima / local minima?
Solution a) From the given factorization, you can just read off the zeros
t = −1, 2, 4, 6. Note that the factor (2 + cos x) is never zero.
b) Put the critical points on a number line, in order, and determine the sign
pattern of f (x). You can do this either by substituting x-values in between
the critical points into f (x), or else by counting the number of negative factors
in f (x) for each of the intervals between critical points. If that number is
even, then f (x) is positive, otherwise negative. Whichever way you do it,
the sign pattern is
−1
2
4
6
x
0
sign of f (x) − | + | − | − | +
This translates directly to f (x) increasing and decreasing as follows.
x
−1
2
4
6
f (x) is & | % | & | & | %
c) you read that table in b) again and see the local minima whenever you
get a pattern &%, local maxima whenever you get a pattern %&.
x
f (x) is &
−1
2
4
6
|
%
|
& | &
|
%
l. min
l. max
l. min