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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 ...
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.
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 .
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 ...
In California and Washington, D.C., for example, some buyers might qualify for up to six figures for their down payment. Grants often come in much smaller amounts, such as $5,000.
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 ...
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
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