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This section explores the concepts of concavity and points of inflection in mathematical functions. It discusses how to determine whether a function is concave upward or downward by analyzing the second derivative across specified intervals. The text provides examples to illustrate finding the open intervals where the graph exhibits different concave behaviors and outlines the definition and significance of points of inflection. Furthermore, it introduces the Second Derivative Test for identifying relative extrema and discusses when to use the First Derivative Test to confirm the results.
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Section 3.4 1st Day Read p. 190.
Test for Concavity • Let f be a function whose second derivative exists on an open interval I. • 1. If f ꞌꞌ(x) > 0 for all x in I, then the graph of f is concave upward in I. • 2. If f ꞌꞌ(x) < 0 for all x in I, then the graph of f is concave downward in I. • Read the paragraph and Note after this theorem on p. 191.
Example 1 • Determine the open intervals on which the graph of • is concave upward or downward.
Then find the 2nd derivative. 1st make sure the graph is continuous.
Use the intervals (-∞, -1), (-1, 1), and (1, ∞) to find where the graph is concave up or concave down.
f is concave up on the intervals (-∞, -1); (1, ∞). • f is concave down on the interval (-1, 1). + − + 1 -1
Concave upward Concave upward Concave downward
Example 2 • Determine the open intervals on which the graph of • is concave upward or downward.
There are no values of x that will make the 2nd derivative equal to 0 but at x = -2 and x = 2 the function is not continuous. Therefore the intervals will be (-∞, -2), (-2, 2), and (2, ∞).
f is concave up on the intervals (-∞, -2); (2, ∞). • f is concave down on the interval (-2, 2). + − + 2 -2
Concave upward Concave upward Concave downward
Definition of Point of Inflection • Let f be a function that is continuous on an open interval, let c be a point in the interval. If the graph of f has a tangent line at this point (c, f(c)), then this point is a point of inflection of the graph if the concavity of f changes from upward to downward (or downward to upward) a the point. • Read the Note on p. 192 and look at the types of points of inflection.
Point of Inflection • If (c, f(c)) is a point of inflection of the graph of f, then either f ꞌꞌ(c) = 0 or f ꞌꞌ does not exist at • x = c.
Example 3 • Determine the points of inflection and discuss the concavity of the graph of
f is concave up on the intervals (-∞, 0); (2, ∞). • f is concave down on the interval (0, 2). + − + 2 0
Concave upward Concave upward Concave downward The points of inflection are at (0, 0) and (2, -16). Read the at last paragraph on p. 193.
2nd Day • Read the 1st paragraph on p. 194.
Second Derivative Test • Let f be a function such that f ꞌ(c) = 0 and the second derivative of f exists on an open interval containing c. • 1. If f ꞌꞌ(c) > 0, then f has a relative minimum at (c, f(c)). • 2. If f ꞌꞌ(c) < 0, then f has a relative maximum at (c, f(c)). • If f ꞌꞌ(c) = 0, the test fails. That is, f may have a relative minimum, relative maximum, or neither. In such cases, you can use the First Derivative Test.
Example • Find the relative extrema for f(x) = -3x5 + 5x3. • Then find the intervals that the graph is concave up and concave down. • Finally find the points of inflection.
1. Find the critical numbers for f by finding the 1st derivative. • f ꞌ(x) = -15x4 + 15x2 • x = -1, 0, 1 • 2. Use the 2nd Derivative Test to find the relative minimum or relative maximum. • f ꞌꞌ(x) = -60x3 + 30x
At x = -1 there is a relative minimum. • At x = 0 we need to use the 1st Derivative Test. • At x = 1 there is a relative maximum. + − = -1 1 0
+ + − − 0
