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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 ∪ {\displaystyle \cup } .
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. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave ...
Specifically, a twice-differentiable function f is concave up if ″ > and concave down if ″ <. Note that if f ( x ) = x 4 {\displaystyle f(x)=x^{4}} , then x = 0 {\displaystyle x=0} has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is ...
The utility function u(c) is defined only up to positive affine transformation – in other words, a constant could be added to the value of u(c) for all c, and/or u(c) could be multiplied by a positive constant factor, without affecting the conclusions. An agent is risk-averse if and only if the utility function is concave.
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 ...
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.
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.
If is concave down in the interval between and , the approximation will be an overestimate (since the derivative is decreasing in that interval). If f {\displaystyle f} is concave up , the approximation will be an underestimate.