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