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Level of detail in writing, sometimes known as level of abstraction, refers to three concepts: the precision in using the right words to form phrases, clauses and sentences; [1] the generality of statements; and the organisational strategy in which authors arrange ideas according to a common topic in the hierarchy of detail.
If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second ...
A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point. It does not, however, provide information about inflection points . Specifically, a twice-differentiable function f is concave up if f ″ ( x ) > 0 {\displaystyle f''(x)>0} and concave down if f ″ ( x ) < 0 ...
The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. [3] [4] [5] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph .
A function f is concave over a convex set if and only if the function −f is a convex function over the set. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
For the graph of a function f of differentiability class C 2 (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 ...
The following are among the properties of log-concave distributions: If a density is log-concave, so is its cumulative distribution function (CDF). If a multivariate density is log-concave, so is the marginal density over any subset of variables. The sum of two independent log-concave random variables is log-concave. This follows from the fact ...
Many optimization problems can be equivalently formulated in this standard form. For example, the problem of maximizing a concave function can be re-formulated equivalently as the problem of minimizing the convex function . The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem.