Maxima & Minima for IIT JEE | askIITians

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Maxima Minima
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Exploring the world of Maxima Minima for
IIT-JEE!!

• The maximum and minimum values of a function
  are together termed as extrema. Pierre de Fermat
  was one of the mathematicians to propose a
  general technique for finding the maxima
  minima.

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The various topics that form a part of maxima and
minima include

• Stationary points
• Turning points
• Local maxima minima
• Ways to identify local maxima minima
• Global maxima minima
• We shall give a brief outline of these topics
  in addition to the tips to master them. Those
  willing to go into the intricacies can refer the
  study material of maxima and minima.

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LOCAL MAXIMA MINIMA

• Mathematically, we can define the local maxima of
  a function f to be a point a such that f(a) gt
  f(a-h) and f(a) gt f(ah), where h gt 0 is a small
  quantity. 
• Similarly, a function f(x) is said to have a
  local minimum at the point a if the value of
  the function at the point x a is less than the
  value at the neighboring points.

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GLOBAL MAXIMA MINIMA

• Global maxima or minima of f(x) in a, b
  refers to the greatest or least value of f(x) in
  a, b. Mathematically, it can be explained as
• The function f(x) has a global maximum at the
  point a in the interval I if
• Similarly, f(x) has a global minimum at the
  point a if f (a) f (x), for all x ? I.
• From the figure given below, it becomes clear
  that while global maxima denotes the maxima of
  the function in the entire interval, local maxima
  denotes the point at which the value of the
  function is maximum as compare to the neighboring
  points.

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What are the various ways of testing the local
maximum and minimum of a function?

• Generally speaking, there are three ways to
  test a function for local maxima/ minima

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Is the topic on maxima and minima important part
of IIT JEE preparation? If yes, then which books
should be referred for it?

• Maxima Minima is an extremely important
  topic of differential calculus. The differential
  calculus portion accounts for 13 of the JEE
  screening and maxima minima is an important
  component of application of derivatives. Some of
  the books which are considered to be best for
  calculus are Arihant differential calculus and
  Das Gupta calculus. These books not only explain
  the concepts in detail but also contain various
  solved and unsolved examples. Various JEE level
  questions are also included in them.

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Illustrations

• Illustration 1
• If f(x) (x2-1)/(x21), for every real number x,
  then the minimum value of f   (IIT JEE 1998) 
• A. does not exist because f is unbounded         
             B. is not attained even though f is
  bounded
• C. is equal to 1                                
                              D. is equal to -1. 
• Solution
• Given f(x) (x2-1)/(x21) 1- 2/(x21)
• This shows that f(x) will be minimum when
  2/(x21) is maximum
• i.e. when (x21) is maximum i.e. at the point x
  0
• Hence, the minimum value of f(x) is f(0) -1.

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• Illustration 2
• Let f(x) x, for 0 lt x 2
•               1, for x 0
• Then at x 0, the function f has (IIT JEE 2000) 
•  A. a local maximum                              
       B. no local maximum
•  C. a local minimum                              
        D. no extremum
• Solution
• First of all, we draw the figure for it.         
                     
• It is clear from the above figure that at the
  point x 0, the function f(x) is not continuous.
• Hence, f has no extremum at x 0.

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Some Interesting Facts

• We can identify the stationary points by looking
  for the points at which dy/dx 0.
• All turning points are stationary points but the
  converse does not hold true, i.e. all stationary
  points are not turning points.
• A maximum (minimum) value of a function may not
  be the greatest (least) value in a finite
  interval.
• A function can have various minimum and maximum
  values and a minimum value may at times exceed
  the maximum value.
• The maximum and minimum values of a function
  always occur alternatively i.e. between every two
  maximum values, there exists a minimum value and
  vice-versa.
• If dy/dx 0 and d2y/dx2gt 0, then that point must
  be a point of minima.
• If dy/dx 0 and d2y/dx2lt 0, then that point must
  be a point of maxima.
• Global maximum and minimum in a, b would always
  occur at critical points of f(x) within a, b or
  at the end pints of the interval provided f is
  continuous in a, b.
• If f(x) is a continuous function on a closed
  bounded interval a,b, then f(x) will have a
  global maximum and a global minimum on a,b.
• If the interval is not bounded or closed, then
  there is no guarantee that a continuous function
  f(x) will have global extrema.
• If f(x) is differentiable on the interval I, then
  every global extremum is a local extremum or an
  endpoint extremum.

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Tips to Study Maxima and Minima

• Maxima minima is an extremely
  simple and an important topic of mathematics
  syllabus of IIT JEE. However, due to lack of
  conceptual clarity, students often get confused
  and commit mistakes in the questions. Here, we
  shall give you some tips which can help you fetch
  perfect scores in this topic
• Do not refer too many books as different books
  use different methods of computing the points of
  extremum.
• Stick to standard books like Arihant Calculus or
  the one by Das Gupta.
• Maxima Minima is a simple topic but consistent
  practice is essential in order to excel in it.
• Various formulas and results must be on your
  fingertips.
• Whenever you are about to attempt a question on
  maxima or minima, read the question carefully to
  identify what it actually demands.
• First of all, you must try to identify the
  stationary points.
• Remember the difference between stationary and
  turning points.
• If possible, try to plot the function on a graph
  as it often helps in reaching at the accurate
  solution easily and quickly.
• In order to identify the local maxima or minima
  of a function, first you must try to apply the
  first derivative test.
• In case you are not able to compute using the
  first derivative test, try to apply the second
  derivative test and if that also fails, then
  proceed toward the nth derivative test.

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Practical Applications of Maxima Minima

• It might sound surprising, but the fact is
  that we can encounter maxima and minima
  anywhere in our daily lives. Some of the
  instances include
• Suppose as a diligent consumer, you wish to
  collect data of your cell phone usage (say) for a
  month. You develop a function representing the
  cell phone usage and then the local maxima/minima
  give you a fair idea of your cell phone usage
  which helps you in choosing the most appropriate
  plan.
• In case we own a company and wish to minimize the
  cost of production, two types of maxima and
  minima can prove useful- absolute maxima minima
  and local maxima and minima.
• A person planning a theme park utilizes the model
  of total revenue as a function of admission
  price. In order to identify the ideal maximum
  price which would be the one that helps in
  fetching the maximum revenue again the concept of
  absolute maximum of this function would be used.
• An actuary also utilizes the concept of local
  minima of the functions in order to identify the
  productive markets and low risk, high yielding
  ventures for his insurance company.
• A NASA engineer working on an innovative space
  shuttle needs to analyze the function that
  calculates the pressure acting on the shuttle at
  a given height. The pressure that the shuttle
  should be capable of withstanding is given by the
  absolute maximum of the function.

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