1 2 Sec 4.3: Concavity and the Second Derivative Test

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Presentation transcript:

1 2 Sec 4.3: Concavity and the Second Derivative Test the curve lies above the tangents the curve lies below the tangents Concavity Test Concavity Test 1 f ’’(x) > 0 for all x in I 2 f ’’(x) < 0 for all x in I f(x) concave Up f(x) concave Down

1 2 Sec 4.3: Concavity and the Second Derivative Test Concavity f ’’(x) > 0 for all x in I 2 f(x) concave Up Concavity f ’’(x) < 0 for all x in I f(x) concave Down

Sec 4.3: Concavity and the Second Derivative Test Example: Study the concavity of the function

Sec 4.3: Concavity and the Second Derivative Test

_ _ + + 1 2 Sec 4.3: Concavity and the Second Derivative Test Inflection point: 1 _ + _ 2 +

Sec 4.3: Concavity and the Second Derivative Test Find all inflection points 7

Sec 4.3: Concavity and the Second Derivative Test Example: Find all inflection points of

Sec 4.3: Concavity and the Second Derivative Test

Sec 4.3: Concavity and the Second Derivative Test

Sec 4.3: Concavity and the Second Derivative Test 1 1 2 2 second Derivative Test: 1 1 2 2 second Derivative Test: 1 1 the test fails. The function ƒ may have a local maximum, a local minimum, or neither. 2 2

Sec 4.3: Concavity and the Second Derivative Test Critical points: Local max Local min

Sec 4.3: Concavity and the Second Derivative Test

Sec 4.3: Concavity and the Second Derivative Test

Sec 4.3: Concavity and the Second Derivative Test How to find local max and local min Find all critical points 1st deriv test 2ed deriv test 1st deriv test No test Graph or others