Untitled Flashcards Set

The First Derivative

• Relative Extrema (f′ = 0):

  • Maximum on f when f′ changes from positive → negative

  • Minimum on f when f′ changes from negative → positive

• Increasing and Decreasing:

  • f is increasing when f′(x) > 0

  • f is decreasing when f′(x) < 0

• Concavity (based on f′):

  • f is concave up when f′(x) is increasing

  • f is concave down when f′(x) is decreasing

• Point of Inflection (POI):

  • There is a POI on f when there is a horizontal tangent on f′

  • (Note from image: When f′(x) goes from increasing to decreasing)

The Second Derivative

• Concavity (based on f″):

  • f is concave up when f″ > 0

  • f is concave down when f″ < 0

• Point of Inflection (POI):

  • f″ = 0 AND

  • f″ changes sign