List of Mathematical Formulas: “Convexity, Concavity and Inflection Points of a Graph

List of formulas and concepts related to convexity, concavity and inflection points of a function graph:

  1. Convexity and concavity of a function:
  • A function ( f(x) ) is convex on an interval if its second derivative ( f”(x) \geq 0 ) on this interval.
  • A function ( f(x) ) is concave on an interval if its second derivative ( f”(x) \leq 0 ) on this interval.
  1. Inflection points:
  • An inflection point is a point ( x = c ) at which the second derivative of a function changes sign. That is, ( f”(c) = 0 ) and ( f”(x) ) changes sign in the vicinity of the point ( c ).
  1. Derivatives:
  • The first derivative: ( f'(x) ) — determines the slope of the function graph.
  • The second derivative: ( f”(x) ) — determines the convexity or concavity of the graph.
  1. Graphical representation:
  • If the graph of a function “looks up” (is convex), then its second derivative is positive.
  • If the graph is “downward-facing” (concave), then its second derivative is negative.

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