Untitled Flashcards Set

Relative Extrema

Locations where the first derivative f′ = 0, indicating potential maxima or minima.

Conditions for Maximum

A maximum occurs on f when the first derivative f′ changes from positive to negative.

Conditions for Minimum

A minimum occurs on f when the first derivative f′ changes from negative to positive.

Increasing Function

A function f is increasing when its first derivative f′(x) > 0.

Decreasing Function

A function f is decreasing when its first derivative f′(x) < 0.

Concavity Up

A function f is concave up when its first derivative f′(x) is increasing.

Concavity Down

A function f is concave down when its first derivative f′(x) is decreasing.

Point of Inflection

There is a point of inflection on f when there is a horizontal tangent on f′.

Concavity from Second Derivative

A function f is concave up when its second derivative f″ > 0 and concave down when f″ < 0.

Conditions for Second Derivative POI

A point of inflection occurs when the second derivative f″ = 0 and changes sign.