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 + _ _ UP _ _ + DOWN + + + UP

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 Example: Study the concavity then find inflection points _ + + + _ _ + + Find the inflection points.

Sec 4. 3: Concavity and the Second Derivative Test F 091 9

Sec 4. 3: Concavity and the Second Derivative Test

Sec 4. 3: Concavity and the Second Derivative Test

Sec 4. 3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH Two methods to find local max and local min First Derivative Test: Second Derivative Test:

Sec 4. 3: Concavity and the Second Derivative Test second Derivative Test: 3 1 2 3 3 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 F 132

Sec 4. 3: Concavity and the Second Derivative Test F 141

Sec 4. 3: Concavity and the Second Derivative Test

Sec 4. 3: Concavity and the Second Derivative Test Which test ? Find all critical points Ex 1 st deriv test 2 ed deriv test ist DNS c t a t con dis c ont a tc 1 st deriv test No test Graph or others