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Application of Derivative:Increasing & Decreasing Function

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Function (mathematics)

A function, in mathematics, associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can also be elements from any given set. An example of a function is f(x) = 2x, a function which associates with every number the number twice as large. Thus 5 is associated with 10, and this is written f (5) = 10.

http://en.wikipedia.org/wiki/Function_(mathematics)

# Monotonic function

In mathematics, a monotonic function (or monotone function) is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

## Monotonicity in calculus and analysis Figure 1. A monotonically increasing function (it is strictly increasing on the left and right while just non-decreasing in the middle.) Figure 2. A monotonically decreasing function.

In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing or non-decreasing), if for all x and y such that x  y one has f(x) ≤ f(y), so f preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing) if, whenever x  y, then f(x) ≥ f(y), so it reverses the order (see Figure 2).

If the order ≤ in the definition of Monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for x not equal to y, either x < y or x > y and so, by Monotonicity, either f(x) < f(y) or f(x) > f(y), thus f(x) is not equal to f(y)).

The terms "non-decreasing" and "non-increasing" are meant to avoid confusion with "strictly increasing" and "strictly decreasing", respectively, but should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing"; see also strict. When functions between discrete sets are considered in combinatorics, it is not always obvious that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, so one finds the terms weakly increasing and weakly decreasing to stress this possibility.

The term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. Notably, this is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences)

A function f(x) is said to be absolutely monotonic over an interval (a, b) if the derivatives of all orders of f are nonnegative at all points on the interval.

http://en.wikipedia.org/wiki/Monotonic_function

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity (conversely, integrating a car's velocity over time yields the distance traveled).

The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. In higher dimensions, the derivative of a function at a point is a linear transformation called the linearization. A closely related notion is the differential of a function.

The process of finding a derivative is called differentiation.

## Computing the derivative

The derivative of a function can, in principle, be computed from the definition by considering the difference quotient, and computing its limit. In practice, once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using rules for obtaining derivatives of more complicated functions from simpler ones.

### Derivatives of elementary functions

Most derivative computations eventually require taking the derivative of some common functions. The following incomplete list gives some of the most frequently used functions of a single real variable and their derivatives.

Derivatives of powers: if where r is any real number, then wherever this function is defined. For example, if f(x) = x1 / 4, then and the derivative function is defined only for positive x, not for x = 0. When r = 0, this rule implies that f′(x) is zero for x ≠ 0, which is almost the constant rule.

Constant rule: if ƒ(x) is constant, then (Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Derivative