5.1: The First Derivative
Given a function f . How the sign of f 0 determine whether f is increasing or decreasing?
Test for Increasing and Decreasing Functions
• If f 0 (x) > 0 on an interval, then f is increasing on that interval.
• If f 0 (x) < 0 on an interval, then f is decreasing on that interval.
Example 1. Given the graph of the derivative f 0 of a function
a) When is f increasing? decreasing?
b) Given f(0)=0, sketch a possible graph of f .
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Given a function f (x). A value x = c is a critical value for f (x) if
1) c is in the domain of the function f (x) and
2) f 0 (c) = 0 or f 0 (c) does not exist.
Constant Sign Theorem: If f (x) is continuous on (a, b) and f (x) 6= 0 for any x in (a, b), then either
f (x) > 0 for all x in (a, b) or f (x) < 0 for all x in (a, b) .
Suppose x = c is a critical value of f (x)
f (c) is a relative (local) maximum if f (x) ≤ f (c) for all x in some interval (a, b) containing c.
f (c) is a relative minimum if f (x) ≥ f (c) for all x in some interval (a, b) containing c.
f (c) is a relative extremum if f (c) is a realtive maximum or a relative minimum.
In plural, relative extrema are all relative maxima and relative minima of f .
Note: All relative extrema occur at critical values, but not all critical values produce relative extrema.
f (c) is an absolute (global) maximum of f (x) if if f (x) ≤ f (c) for all x in the domain.
f (c) is an absolute minimum of f (x) if if f (x) ≥ f (c) for all x in the domain.
f (c) is an absolute extremum if f (c) is an absolute maximum or an absolute minimum.
In plural, extrema are the maximum and minimum of f .
Example 2. (Stewart) Given is the graph of a function. For each of the numbers a, b, c, d, e, r, s, and
t, state whether the function has an absolute maximum or minimum, a relative maximum or minimum,
or neither a maximum nor a minimum.
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The First Derivative Test: Suppose that c is a critical point of a continuous function f .
1) If the sign of f 0 changes from + to - at c, then f has a relative maximum at c.
2) If the sign of f 0 changes from - to + at c, then f has a relative minimum at c.
3) If f 0 does not change sign at c, then f has no relative extrema at c.
Example 3. Let f (x) = x3 − 3x2 + 1.
a) Find the critical values of f .
b) Find x where f is increasing and where f is decreasing.
c) Find the relative extremum of the function f .
d) Use the above information, sketch a graph.
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Example 4. Let f (t) =
(t − 1)2
.
t+2
a) Find the critical values of f .
b) Find t where f is increasing and where f is decreasing.
c) Find the relative extremum of the function f .
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Example 5. Let f (s) = x1/3 (100 − x).
a) Find the critical values of f .
b) Find s where f is increasing and where f is decreasing.
c) Find the relative extremum of the function f .
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