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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.
printf is a C standard library function that formats text and writes it to standard output. The name, printf is short for print formatted where print refers to output to a printer although the functions are not limited to printer output. The standard library provides many other similar functions that form a family of printf-like functions.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
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 ...
When f is a function of several variables, it is common to use "∂", a stylized cursive lower-case d, rather than "D". As above, the subscripts denote the derivatives that are being taken. For example, the second partial derivatives of a function f(x, y) are: [6]
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the Gaussian function f(x) = exp(−x 2 /2) which is log-concave since log f(x) = −x 2 /2 is a concave function of x. But f is not concave since the second derivative is positive for | x | > 1:
Modifier Letter Prime U+02BA ʺ 698 Modifier Letter Double Prime U+02BB ʻ 699 Modifier Letter Turned Comma 0356 in Sami: U+02BC ʼ 700 Modifier Letter Apostrophe: 0357 in ISO/IEC 8859-7: U+02BD ʽ 701 Modifier Letter Reversed Comma 0358 U+02BE ʾ 702 Modifier Letter Right Half Ring · U+02BF ʿ 703 Modifier Letter Left Half Ring U+02C0 ˀ 704
f and g are lower semi-continuous and ( ) where is the algebraic interior and , where h is some function, is the set {: < +}, or A dom f ∩ cont g ≠ ∅ {\displaystyle A\operatorname {dom} f\cap \operatorname {cont} g\neq \emptyset } where cont {\displaystyle \operatorname {cont} } are the points where the function is ...