Concave function in the context of Derivative test

An ogee (/oʊˈdʒiː/ /ˈoʊdʒiː/) is an object, element, or curve—often seen in architecture and building trades—that has a serpentine- or extended S-shape (sigmoid). Ogees consist of a "double curve", the combination of two semicircular curves or arcs that, as a result of a point of inflection from concave to convex or vice versa, have ends of the overall curve that point in opposite directions (and have tangents that are approximately parallel).

First seen in textiles in the 12th century, the use of ogee elements—in particular, in the design of arches—has been said to characterise various Gothic and Gothic Revival architectural styles. The shape has many such uses in architecture from those periods to the present day, including in the ogee arch in these architectural styles, where two ogees oriented as mirror images compose the sides of the arch, and in decorative molding designs, where single ogees are common profiles (see opening image). The term is also used in marine construction, particularly in shipbuilding, where ogee curves are used in hull design to improve hydrodynamics.The word was sometimes abbreviated as o-g as early as the 18th century, and in millwork trades associated with building construction, ogee is still sometimes written similarly (e.g., as O.G.).

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

For the graph of a function f of differentiability class C (its first derivative f', and its second derivative f'', exist and are continuous), the condition f'' = 0 can also be used to find an inflection point since a point of f'' = 0 must be passed to change f'' from a positive value (concave upward) to a negative value (concave downward) or vice versa as f'' is continuous; an inflection point of the curve is where f'' = 0 and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.

In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on the graph of the function between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph is shaped like a cup ${\displaystyle \cup }$ (or a straight line like a linear function), while a concave function's graph is shaped like a cap ${\displaystyle \cap }$.

A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function ${\displaystyle f(x)=cx}$ (where ${\displaystyle c}$ is a real number), a quadratic function ${\displaystyle cx^{2}}$ (${\displaystyle c}$ as a nonnegative real number) and an exponential function ${\displaystyle ce^{x}}$ (${\displaystyle c}$ as a nonnegative real number).