Application of Derivative – Maxima and Minima  Mathematics
The Concept of derivative can be used to find the maximum and minimum value of the given function. We know that information about and gradient or slope can be derived from the derivative of a function. We try to find a point which has zero gradients then locate maximum and minimum value near it. It is of use because it can be used to maximize profit for a given curve or minimizing loses or costs depending on the area of its usage.
Note: If f(x)
is a continuous function, then for every continuous function on a closed interval has a maximum and a minimum value.
 Let f be a realvalued function and let a be an interior point in the domain of f. Then
 ‘a’ is called a point of local maxima if there is an h > 0 such that f(a) ≥f(x), for all x in (a – h, a + h), x≠a The value f(a) is called the local maximum value of f.
 ‘a’ is called a point of local minima if there is an h > 0 such that f(a) ≥ f(x), for all x in (a – h, a + h) The value f(a) is called the local minimum value of f
Note: Let f be a function defined on an open interval I. Suppose c ∈ I be any point. If f has a local maxima or a local minima at x = c, then either f^{‘}(c) = 0 or f is not differentiable at c.
 Steps to find maxima and minima –

 First derivative test
 If changes it’s sign from positive to negative then the point c at which it happens is local maxima.
 If changes it’s sign from negative to positive then the point c at which it happens is local minima.
 If does not change it’s sign as x increases through c then the point is point of inflection.
In the above diagram C_{3} is global maxima and C_{4} local minima. 
 Second derivative test
 Find values of x for which , these points are called critical points.
 Find and put the values of x which was found above, if
 then the point is minima
 then the point is minima
 then we can not say anything, now we have to use first derivative to check whether the point is point of inflection, local minima or local maxima.
Stationary Point:
A point on which the tangent to the graph is horizontal is known as a stationary point, i.e the point at which .
Note: If you have to find the maximum and minimum value of a function in a closed interval then find all the critical points by equating , then find the value of f(x) on all the points in the given interval [a, b].
 Related Gate Questions:
 Gate CS 2012
Recommended Posts:
 Mathematics  Set Operations (Set theory)
 Mathematics  The Pigeonhole Principle
 Mathematics  Matrix Introduction
 Mathematics  Introduction to Propositional Logic  Set 1
 Mathematics  Propositional Equivalences
 Mathematics  Predicates and Quantifiers  Set 1
 Mathematics  L U Decomposition of a System of Linear Equations
 Mathematics  Introduction of Set theory
 Mathematics  Mean, Variance and Standard Deviation
 Mathematics  Conditional Probability
 Mathematics  Sum of squares of even and odd natural numbers
 Mathematics  Eigen Values and Eigen Vectors
 Mathematics  Rolle's Mean Value Theorem
 Mathematics  Lagrange's Mean Value Theorem
 Mathematics  Random Variables
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.