dönüümlerin

Dönümlerin is a Turkish term that translates roughly to “the turning points.” In Turkish-language usage, it refers to points at which a process changes direction, or where a function changes monotonicity. The word is built from dönüm (turning point) with plural and grammatical suffixes, and dönümlerin specifically denotes the genitive plural form used when saying “the turning points of …” in a sentence.

In mathematics, dönümlerin typically denote turning points of a function, i.e., points where the function changes from increasing to decreasing or from decreasing to increasing. These are often critical points where the derivative f′(x) equals zero or is undefined. The standard analytical tests to identify them include the first derivative test (examining sign changes of f′), the second derivative test (f″(x) > 0 for a local minimum, f″(x) < 0 for a local maximum), and higher-order tests when needed. It is important to distinguish turning points from inflection points, the latter signaling a change in concavity rather than a change of monotonicity.

Beyond calculus, the term appears in physics and history with related meanings: a turning point can describe a moment when motion reverses direction in a trajectory, or a milestone that alters the course of events in a historical narrative. In Turkish, the broader phrase dönüm noktası is widely used for such milestones.

Examples commonly cited include f(x) = x^2, which has a turning point at x = 0 (local minimum), and f(x) = -x^2, which has a turning point at x = 0 (local maximum). See also turning point, local maximum, local minimum, critical point, and inflection point.