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3.3:
st
1 Derivative Test
By: Ted feat. G$ and Jameth
Increasing vs. Decreasing
 A function f is increasing on an interval if for any
two numbers x1 and x2 in the interval, x1 < x2
implies f(x1) < f(x2).
 A function f is decreasing on an interval if for
any two numbers x1 and x2 in the interval, x1 < x2
implies f(x1) > f(x2).
 In other words, a function is increasing if, as x
moves to the right, its graph moves up and
decreasing if the graph moves down.
Increasing vs. Decreasing
 Theorem 3.5:
Let f be a function that is continuous on the
closed interval [a, b] and differentiable on the
open interval (a, b).
• 1. If f’(x) > 0 for all x in (a, b), then f is
increasing on [a, b].
• 2. If f’(x) < 0 for all x in (a, b), then f is
decreasing on [a, b].
• 3. If f’(x) = 0 for all x in (a, b), then f is
constant on [a, b].
Increasing vs. Decreasing
 In simpler terms, Theorem 3.5 states…
 1. A positive derivative implies that the function
is increasing.
 2. A negative derivative implies that the
function is decreasing.
 3. A zero derivative implies that the function is
constant.
Example: f(x) =
3
x
• Step 1: Find the derivative of f(x).
Answer: f’(x)=3x2 – 3x
–
2
1.5x
Example (continued)
 Step 2: Graph the derivative.
 Step 3: Analyze the graph of
the derivative.
On the intervals (-∞, 0)
and (1, ∞), f’(x) is positive
and so f(x) must be increasing
on those intervals.
On the interval (0, 1),
however, f’(x) is negative and
so f(x) must be decreasing on
that interval.
Example (continued)
•In this comparison of the
top graph f(x), and the
bottom graph f’(x), not
only does Theorem 3.5
hold true, but, in addition,
at the two values where
f’(x) intersects with the xaxis, there are a relative
maximum and minimum
at the corresponding
value on f(x).
st
1
Derivative Test
 Let c be a critical number of a function f that is
continuous on an open interval containing c. If f is
differentiable on the interval, except possibly at c,
then f(c) can be classified as follows:
1. If f’(x) changes from negative to positive at c,
then f(c) is a relative minimum of f.
2. If f’(x) changes from positive to negative at c,
then f(c) is a relative maximum of f.
3. If f’(x) does not change sign at c, then f(c) is
neither a relative minimum nor a relative
maximum.
Example: f(x) =
2
x
– 6x
 Find the critical numbers for the function.
 Step 1: Take the derivative… f’(x) = 2x – 6.
 Step 2: Set f’(x) equal to zero and then solve for x
to find the critical number(s), if any… 2x – 6 = 0,
so x = 3. [Note: The graph of f’(x) crosses the xaxis when f’(x) = 0. So the critical numbers of f(x)
correspond to x-intercepts of f’(x).]
Example (continued)
 Step 3: Clearly, there is a
critical number at x = 3.
To find out if this point is
a relative minimum or
maximum, look back at
the graph of f(x).
 Step 4: Based on the
graph, we can see that
the critical number is a
relative minimum.
Example (continued)
 Note: If you are not given a graph and/or cannot
generate one, you can determine if it’s a relative
minimum or maximum by plugging in values
for x into the derivative function that are very
close to the critical number.
 In the previous example, the critical number was
at x = 3. So plug in 2.9 and 3.1 for x into the
derivative equation.
f’(2.9) = -.2
f’(3.1) = .2
Since it changed from negative to positive, the
critical point is a relative minimum of f(x).
Made by…
Ted “the goods” Murphy
Michael G$ Goepfrich
Jameth O’Brien