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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 .
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.
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 ...
State or local housing authorities, a non-profit organization, or lender usually set the requirements and conditions for the DPA program. Some programs require you or your loan officer to take a short course on Down Payment Assistance for first time home buyers. [2] Homebuyer assistance programs vary by state and can amount to up to three ...
Finite-dimensional real vector spaces with (pseudo-)metrics are classified up to signature, a coordinate-free property which is well-defined by Sylvester's law of inertia. Possible metrics on real space are indexed by signature (,). This is a metric associated to = + dimensional real space.
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 ...
This can also be seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it. Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, the ...
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 ...