1st and 2nd Derivative Test

f is increasing

f’ +

f is decreasing

f’ -

f is concave up

f’’ +

f is concave down

f’’ -

Horizontal Tangent Line or Sharp Turn

f’ = 0 or f’ DNE

The point that f changes concavity

f’’ = 0 or DNE AND f’’ changes signs

1st derivative definition of the relative max of f (mountain)

f’ = 0 or DNE AND f’ changes sign from + to -

1st derivative definition of the relative min of f (valley)

f’ = 0 or DNE AND f’ changes signs from - to +

2nd derivative definition of the relative max of f (mountain)

f’ = 0 or DNE AND f’’ -

2nd derivative definition of the relative min of f (valley)

f’ = 0 or DNE AND f’’ +