Transcript Section 4.1 Using First and Second Derivatives

Section 4.1
Using First and Second Derivatives
• Let’s see what we remember about derivatives
of a function and its graph
– If f’ > 0 on an interval than f is
• Increasing
– If f’ < 0 on an interval than f is
• Decreasing
– If f’’ > 0 on an interval than the graph of f is
• Concave up
– If f’’ < 0 on an interval than the graph of f is
• Concave down
• Consider the function
f ( x)  x  6x  9x  5
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• Let’s find where it is increasing, decreasing,
concave up, and concave down algebraically
and then check that with the graph
Critical Points
• For any function f, a point p in the domain of f
where f’(p) = 0 or f’(p) is undefined is called a
critical point of the function
– The critical value of f is the function value, f(p)
where p is the critical point
– Critical points are used to determine relative
extrema
Relative Extrema
• f has a local maximum at x = p if f(p) is equal to
or larger than all other f values near p
– If p is a critical point and f’ changes from positive to
negative at p, then f has a local maximum at p
• f has a local minimum at x = p if f(p) is equal to
or smaller than all other f values near p
– If p is a critical point and f’ changes from negative to
positive at p, then f has a local maximum at p
• Since in the previous cases we were using the
first derivative, we were using the first
derivative test to check for relative extrema
The Second Derivative Test for Relative Extrema
Suppose f’(p) = 0 and thus p is a critical point of
f
• If f’’(p) < 0 then
– f has a local maximum at p
• If f’’(p) > 0 then
– f has a local minimum at p
• If f’’(p) = 0 then
– The test tells us nothing
• Places where the graph switches concavity are
called inflection points
• How can we identify inflection points?
– Where the second derivative is zero or undefined
Example
• For the following function find where it is
increasing, decreasing, concave up, and
concave down algebraically and identify any
relative extrema and/or inflection points
f ( x)  x 15x  10
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