Explanation:
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.
http://en.wikipedia.org/wiki/Maxima_and_minima
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)
http://en.wikipedia.org/wiki/First_derivative_test
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 0" *> 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 0" *>
http://en.wikipedia.org/wiki/Second_derivative_test
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)
= x^{1 / 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)
http://en.wikipedia.org/wiki/Derivative
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