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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 } .
In accounting, a down payment (also called a deposit in British English) is an initial up-front partial payment for the purchase of expensive goods or services such as a car or a house. It is usually paid in cash or equivalent at the time of finalizing the transaction .
The operating systems the software can run on natively (without emulation).Android and iOS apps can be optimized for Chromebooks and iPads which run the operating systems ChromeOS and iPadOS respectively, the operating optimizations include things like multitasking capabilities, large and multi-display support, better keyboard and mouse support.
risk neutral – if they are indifferent between the bet and a certain $50 payment. risk loving (or risk seeking) – if they would accept the bet even when the guaranteed payment is more than $50 (for example, $60). The average payoff of the gamble, known as its expected value, is $50.
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 ...
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 ...
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 ...