Concavity and Second Derivative Test Lesson 4.4 Concavity • Concave UP • Concave DOWN • Inflection point: Where concavity changes
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Transcript Concavity and Second Derivative Test Lesson 4.4 Concavity • Concave UP • Concave DOWN • Inflection point: Where concavity changes
Concavity and Second
Derivative Test
Lesson 4.4
Concavity
• Concave UP
• Concave DOWN
• Inflection point:
Where concavity
changes
Inflection Point
• Consider the slope as curve changes
through concave up to concave down
At inflection point
slope reaches
maximum
positive value
Slope starts
negative
Slope
Becomes
Slopebecomes
becomes
zero
less negative (horizontal)
positive, then
more positive
After inflection
point, slope
becomes less
positive
Graph of the slope
Inflection Point
• What could you say about the slope function
when the original function has an inflection
point
• Slope function has a maximum (or minimum
• Thus second derivative = 0
Graph of the slope
Second Derivative
• This is really the rate of change of the slope
• When the original function has a relative
minimum
Slope is increasing (left to right) and goes
through zero
Second derivative is positive
Original function is concave up
Second Derivative
• When the original function has a relative
maximum
The slope is decreasing (left to right) and goes
through zero
The second derivative is negative
The original function is
concave down
View Geogebra
Demo
Second Derivative
• If the second derivative f ’’(x) = 0
The slope is neither increasing nor decreasing
• If f ’’(x) = 0 at the same place f ’(x) = 0
The 2nd derivative test fails
You cannot tell what the function is doing
f ''( x) 12x
2
f ( x) x
Not an inflection
point
4
Example
• Consider
f ( x) x 3x 4
3
• Determine f ‘(x) and f ’’(x) and when they are
zero
f '( x) 3x 3 0 when x 1
2
f ''( x) 6 x 0 when x 0
Example
f ( x) x 3x 4
3
f ’(x) = 0, f’’(x) > 0,
this is concave up, a
relative minimum
f ‘(x)
f(x)
f ‘(x) = 0, f ‘’(x) < 0
this is concave
down, a maximum
f ‘’(x)
f ‘’(x) = 0 this is an
inflection point
Example
• Try
f ( x) x 1
2
• f ’(x) = ?
• f ’’(x) = ?
• Where are relative max, min, inflection point?
Algorithm for Curve Sketching
• Determine critical points
Places where f ‘(x) = 0
• Plot these points on f(x)
• Use second derivative f’’(x) = 0
Determine concavity, inflection points
• Use x = 0 (y intercept)
• Find f(x) = 0 (x intercepts)
• Sketch
Assignment
• Lesson 4.4
• Page 235
• Exercises 5, 12, 19, 26, 33,
40, 47, 54, 61, 68, 73, 83, 91,
95, 96