Section 3.4: Concavity and the second Derivative Test. Find any points of inflection of the graph of a function.
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1 Unit 3: Applications o Dierentiation Section 3.4: Concavity and the second Derivative Test Determine intervals on which a unction is concave upward or concave downward. Find any points o inlection o the graph o a unction. Apply the Second Derivative Test to ind relative etrema o a unction.
2 Deinition o Concavity Let be dierentiable on an open interval I. the graph o is concave upward on I i is increasing on the interval and concave downward on I i is decreasing on the interval. Eample 1: Concavity Concave downward, is decreasing. Concave upward, is increasing.
3 Theorem 3.7 Test or Concavity Let be a unction whose second derivative eist on an open interval I. 1. I () > 0 or all in I, then the graph o is concave upward in I.. I () < 0 or all in I, then the graph is concave downward in I.
4 Eample : Determine Concavity Determine the open interval on which the graph o is concave upward or downward ' 1 1 ' '' Interval Test value Sign o '' Conclusion '' 3 0 '' 0 0 concave upward 0 3 concave downward '' 3 0 concave upward
5 () > 0 is concave upward () < 0 is concave downward Eample 3: Determine Concavity Determine the open interval on which the graph o is concave upward or downward. ' '' Interval Test value Sign o '' Conclusion '' 4 0 '' 0 0 concave upward 0 4 concave downward 9 '' 4 0 concave upward 90 3
6 Deinition o point o inlection Let be a unction that is continuous on an open interval and let c be a point in the interval. I the graph o has a tangent line at this point ( c, (c)), then this point is a point o inlection o the graph o i the concavity o changes rom upward to downward (or downward to upward) at the point. Eample 3: Point o Inlection
7 Deinition o point o inlection Let be a unction that is continuous on an open interval and let c be a point in the interval. I the graph o has a tangent line at this point ( c, (c)), then this point is a point o inlection o the graph o i the concavity o changes rom upward to downward (or downward to upward) at the point. Eample 4: Point o Inlection
8 Theorem 3.8 Point o Inlection I (c, (c)) is a point o inlection o the graph o, then either (c) = 0 or does not eist at = c. Eample 5: Finding Points o Inlection Determine the point o inlection and discuss the concavity o ' '' 1 6 Interval Test value Sign o '' '' 1 0 '' Conclusion concave concave downward upward Point o Inlection 1, 3
9 HW 1 Section 3.4 Page 195 E:(18) 1-10, 11-5 O
10 Theorem 3.9 Second Derivative Test Let be a unction such that (c) = 0 and the second derivative o eist on an open interval containing c. 1. I (c) > 0, then has a relative minimum at (c, (c)).. I (c) < 0, then has a relative maimum at (c, (c)). I (c) = 0, the test ails that is, may have a relative maimum, a relative minimum, or neither. In such cases you can use the First Derivative Test.
11 '' c 0 '' c 0 Concave downward Concave upward c I (c) = 0 and > 0, (c) is a relative minimum c I (c) = 0 and < 0, (c) is a relative maimum
12 (c) > 0, has a r. minimum at (c, (c)) (c) < 0, has a r. maimum at (c, (c)). Eample 6: Using the second Derivative Test 3 Find the relative etrema or 9 7 derivative test where applicable. use the second ' c 3 '' 6 18 Interval Test value Sign o ' Conclusion ' 1 0 ' 4 0 () > 0 or all, thereore there are not relative etrema '' 3 0 test ails
13 (c) > 0, has a r. minimum at (c, (c)) (c) < 0, has a r. maimum at (c, (c)). Eample 7: Using the second Derivative Test ' cos sin cos sin 0 cos c sin 5, 4 4 '' sin cos sin cos Find the relative etrema or derivative test where applicable. '' '' 0 4, 4 5, 4 use the second is a relative maimum is a relative minimum
14 (c) > 0, has a r. minimum at (c, (c)) (c) < 0, has a r. maimum at (c, (c)). Eample 8: Sketch the graph Sketch the graph o a unctions having the given characteristics. 0 0 ' 0 i 1 ' 1 0 ' 0 i 1 '' 0
15 Eample 9a: AP Eam Type Question Let be the unction deined by k ln or > 0, where k is a positive constant. k 1 ' 1 3 '' k 4 (a)for what value o the constant k does have a critical number at the point = 1? For this value k, determine whether has a relative minimum, relative maimum, or neither at = 1. justiy your answer. k 1 k 1 k '1 0 1 k When k = (1) = 0 and (1) > 0. So by the Second Derivative Test has a relative minimum at = 1.
16 Eample 9b: AP Eam Type Question Let be the unction deined by k ln or > 0, where k is a positive constant. k 1 ' 1 3 '' k 4 (b) For a certain value o the constant k, the graph o has a point o inlection on the -ais. Find this value o k. A this inlection point () = 0 and '' 0 k k k ln 0 Thereore, ln 4 1 k ln () = 0. k ln 4 e k e k 3 ln k 4 3 4
17 (c) > 0, has a r. minimum at (c, (c)) (c) < 0, has a r. maimum at (c, (c)). Eample 10: Second Derivative Test Find a, b, c, and d such that the cubic satisy the given conditions. ' 3a b c relative maimum (, 4) '' 6a b 3 a b c d 8a 4b c d 4 ' 1a 4b c 0 relative minimum (4, ) Inlection point (3, 3) 4 64a 16b 4c d ' 4 48a 8b c 0 '' 3 18a b 0
18 Eample 10: Second Derivative Test Find a, b, c, and d such that the cubic satisy the given conditions. 8a 4b c d 4 ' 1a 4b c a 16b 4c d ' 4 48a 8b c 0 '' 3 18a b 0 a 3 a b c d , 9 b, c 1, d
19 HW Section 3.4 Page 195 E:(18) 7-39 O, 45, O, 61-63, 68-69
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