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Application of Derivative:Finding Maxima or Minima

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Maxima and minima


In mathematics, the maximum and minimum (plural: maxima and minima) of a function, known collectively as extrema (singular: extremum), are the largest and smallest value that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).

More generally, the maximum and minimum of a set (as defined in set theory) are the greatest and least element in the set. Unbounded infinite sets such as the set of real numbers have no minimum and maximum.


Local and global maxima and minima for cos(3πx)/x, 0.1≤x≤1.1


Analytical definition

A real-valued function f defined on a real line is said to have a local (or relative) maximum point at the point x, if there exists some ε > 0 such that f(x) ≥ f(x) when |x  x| < ε. The value of the function at this point is called maximum of the function. Similarly, a function has a local minimum point at x, if f(x) ≤ f(x) when |x  x| < ε. The value of the function at this point is called minimum of the function.







First derivative test


In calculus, the first derivative test uses the first derivative of a function to determine whether a given critical point of a function is a local maximum, a local minimum, or neither.


Precise statement of first derivative test

The first derivative test depends on the "increasing-decreasing test", which is itself ultimately a consequence of the mean value theorem.

Suppose f is a real-valued function of a real variable defined on some interval containing the critical point x. Further suppose that f is continuous at x and differentiable on some open interval containing x, except possibly at x itself.

1.  If there exists a positive number r such that for every y in (x - r, x) we have f'(y) ≥ 0, and for every y in (x, x + r) we have f'(y) ≤ 0, then f has a local maximum at x.

2.  If there exists a positive number r such that for every y in (x - r, x) we have f'(y) ≤ 0, and for every y in (x, x + r) we have f'(y) ≥ 0, then f has a local minimum at x.

3.  If there exists a positive number r such that for every y in (x - r, x)  (x, x + r) we have f'(y) > 0, or if there exists a positive number r such that for every y in (x - r, x)  (x, x + r) we have f'(y) < 0, then f has neither a local maximum nor a local minimum at x.

4.  If none of the above conditions hold, then the test fails. (Such a condition is not vacuous; there are functions that satisfy none of the first three conditions.)

Again, corresponding to the comments in the section on monotonicity properties, note that in the first two cases, the inequality is not required to be strict, while in the third case, strict inequality is required.

(Our solved example in mathguru.com uses this concept)




Second derivative test

In calculus, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum using the value of the second derivative at the point.

The test states: If the function f is twice differentiable at a stationary point x, meaning that ,then

1.  If  then  has a local maximum at .

2.  If  then  has a local minimum at .

3.  If , the second derivative test says nothing about the point , a possible inflection point.

(Our solved example in mathguru.com uses this concept)

In the last case, although the function may have a local maximum or minimum at x, because the function is sufficiently "flat" (i.e. ) the extremum is rendered undetected by the second derivative. In this case one has to examine the third derivative. The point at which  is an inflection point if concavity changes on either side of it. For example, (0,0) is an inflection point on  because , and  and 






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).



 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.

1.       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.

2.  Constant rule:if ƒ(x) is constant, then

(Our solved example in mathguru.com uses this concept)




The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.