First derivative test
Encyclopedia
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.
properties of a function just to the left and right of a given point in its domain. If the function "switches" from increasing to decreasing at the point, then close to that point, it will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then close to that point, it will achieve a least value at that point. If the function fails to "switch", and remains increasing or remains decreasing, then no highest or least value is achieved.
The general idea of examining monotonicity does not depend on calculus. However, calculus is introduced because there are sufficient conditions
that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.
Note that in the first two cases, f is not required to be strictly increasing or strictly decreasing to the left or right of x, while in the last two cases, f is required to be strictly increasing or strictly decreasing. The reason is that in the definition of local maximum and minimum, the inequality is not required to be strict: e.g. every value of a constant function is considered both a local maximum and a local minimum.
.
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.
Again, corresponding to the comments in the section on monotonicity properites, note that in the first two cases, the inequality is not required to be strict, while in the third case, strict inequality is required.
s in physics in engineering. In conjunction with the extreme value theorem
, it can be used to find the absolute maximum and minimum of a real-valued function defined on a closed, bounded interval. In conjunction with other information such as concavity, inflection points, and asymptotes, it can be used to sketch the graph of a function.
In Mathematics textbooks for Secondary level students, the first derivative test is used to find the absolute maximum and minimum of a function. It can by found by considering f `(x) = o and finding critical points(a and b). Then f(a), f(b) and f(Intervals if any is given) is found. The largest of the lot is considered as absolute maximum and the least is absolute minimum.
For the concept of Local maximum and Minimum, the second derivative test can be used to get the result in a quick and precise manner when compared to the first derivative test.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, the first derivative test uses the first derivative
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...
of a function to determine whether a given critical point
Stationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
of a function is a local maximum, a local minimum, or neither.
Intuitive explanation
The idea behind the first derivative test is to examine the monotonicMonotonic function
In mathematics, a monotonic 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....
properties of a function just to the left and right of a given point in its domain. If the function "switches" from increasing to decreasing at the point, then close to that point, it will achieve a highest value at that point. Similarly, if the function "switches" from decreasing to increasing at the point, then close to that point, it will achieve a least value at that point. If the function fails to "switch", and remains increasing or remains decreasing, then no highest or least value is achieved.
The general idea of examining monotonicity does not depend on calculus. However, calculus is introduced because there are sufficient conditions
Necessary and sufficient conditions
In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...
that guarantee the monotonicity properties above, and these conditions apply to the vast majority of functions one would encounter.
Precise statement of monotonicity properties
Stated precisely, suppose f is a real-valued function of a real variable, defined on some interval containing the point x.- If there exists a positive number r such that f is non-decreasing on (x - r, x] and non-increasing on [x, x + r), then f has a local maximum at x.
- If there exists a positive number r such that f is non-increasing on (x - r, x] and non-decreasing on [x, x + r), then f has a local minimum at x.
- If there exists a positive number r such that f is strictly increasing on (x - r, x] and strictly increasing on [x, x + r), then f is strictly increasing on (x - r, x + r) and does not have a local maximum or minimum at x.
- If there exists a positive number r such that f is strictly decreasing on (x - r, x] and strictly decreasing on [x, x + r), then f is strictly decreasing on (x - r, x + r) and does not have a local maximum or minimum at x.
Note that in the first two cases, f is not required to be strictly increasing or strictly decreasing to the left or right of x, while in the last two cases, f is required to be strictly increasing or strictly decreasing. The reason is that in the definition of local maximum and minimum, the inequality is not required to be strict: e.g. every value of a constant function is considered both a local maximum and a local minimum.
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 theoremMean value theorem
In calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
.
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
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
at x and differentiable
Differentiable function
In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...
on some open interval containing x, except possibly at x itself.
- 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.
- 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.
- 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.
- If none of the above conditions hold, then the test fails. (Such a condition is not vacuousVacuous truthA vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If A then B" when in fact A is inherently false. For example, the statement "all cell phones in the room are turned off" may be true...
; there are functions that satisfy none of the first three conditions.)
Again, corresponding to the comments in the section on monotonicity properites, note that in the first two cases, the inequality is not required to be strict, while in the third case, strict inequality is required.
Applications
The first derivative test is helpful in solving optimization problemOptimization problem
In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete...
s in physics in engineering. In conjunction with the extreme value theorem
Extreme value theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous in the closed and bounded interval [a,b], then f must attain its maximum and minimum value, each at least once...
, it can be used to find the absolute maximum and minimum of a real-valued function defined on a closed, bounded interval. In conjunction with other information such as concavity, inflection points, and asymptotes, it can be used to sketch the graph of a function.
In Mathematics textbooks for Secondary level students, the first derivative test is used to find the absolute maximum and minimum of a function. It can by found by considering f `(x) = o and finding critical points(a and b). Then f(a), f(b) and f(Intervals if any is given) is found. The largest of the lot is considered as absolute maximum and the least is absolute minimum.
For the concept of Local maximum and Minimum, the second derivative test can be used to get the result in a quick and precise manner when compared to the first derivative test.
See also
- Fermat's theorem (stationary points)Fermat's theorem (stationary points)In mathematics, Fermat's theorem is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point...
- Maxima and minimaMaxima and minimaIn mathematics, the maximum and minimum of a function, known collectively as extrema , are the largest and smallest value that the function takes at a point either within a given neighborhood or on the function domain in its entirety .More generally, the...
- Second derivative testSecond derivative testIn 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....
- Higher-order derivative test
- Karush–Kuhn–Tucker conditions
- Phase linePhase line (mathematics)In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, dy/dt = ƒ...
– virtually identical diagram, used in the study of ordinary differential equations