Sec 4 3 Concavity and the Second Derivative

Sec 4. 3: Concavity and the Second Derivative Test the curve lies above the tangents Concavity Test 1 f ’’(x) > 0 for all x in I f(x) concave Up the curve lies below the tangents Concavity Test 2 f ’’(x) < 0 for all x in I f(x) concave Down

Sec 4. 3: Concavity and the Second Derivative Test Concavity 1 f ’’(x) > 0 for all x in I f(x) concave Up 2 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 F 092

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

Sec 4. 3: Concavity and the Second Derivative Test Example: Find all inflection points of

Sec 4. 3: Concavity and the Second Derivative Test F 091 8

Sec 4. 3: Concavity and the Second Derivative Test

Sec 4. 3: Concavity and the Second Derivative Test second Derivative Test: 1 2 the test fails. The function ƒ may have a local maximum, a local minimum, or neither.

Sec 4. 3: Concavity and the Second Derivative Test F 121 Critical points: Local max Local min

Sec 4. 3: Concavity and the Second Derivative Test F 081

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 1 st deriv test 2 ed deriv test 1 st deriv test No test Graph or others