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Concept of local maximum and local minimum

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Let y = f(x) be a function defined at x = a and also in the vicinity of the point x = a. Then, f(x) is said to have a local maximum at x = a, if the value of the function at x = a is greater than the value of the function at the neighboring points of x = a.
 
Mathematically, f(a) > f(ah) and f(a) > f(a + h), where h > 0. Similarly, f(x) is said to have a local minimum at x = a, if the value of the function at x = a is less than the value of the function at the neighboring points of x = a. Mathematically, f(a) < f(ah) and f(a) < f(a + h), where h > 0.
 
A local maximum or a local minimum is also called a local extremum.

Test for local maximum/minimum

We have two cases to consider:
  1. Test for local maximum/minimum at x = a if f(x) is differentiable at x = a: If f(x) is differentiable at x = a and if it is a critical point of the function (i.e., f′(a) = 0) then we have the following three tests to decide whether f(x) has a local maximum or local minimum or neither at x = a.
    1. First derivative test: If f′(a) = 0 and f′(x) changes its sign while passing through the point x = a, then
      • f(x) would have a local maximum at x = a if f′(a – 0) > 0 and f′(a + 0) < 0. It means that f′(x) should change its sign from positive the to negative.
      • f(x) would have local minimum at x = a if f′(a – 0) < 0 and f′(a + 0) > 0. It means that f′(x) should change its sign from the negative and positive.
      • If f(x) does not change its sign while passing through x = a, then f(x) would have neither a maximum nor a minimum at x = a.
    2. Second derivative test: This test is basically the mathematical representation of the first derivative test. It simply says that
      • If f′(a) = 0 and f′′(a) < 0, then f(x) would have a local maximum at x = a.
      • If f′(a) = 0 and f′′(a) > 0, then f(x) would have a local minimum at x = a.
      • If f′(a) = 0 and f′′(a) = 0, then this test fails and the existence of a local maximum/minimum at x = a is decided on the basis of the nth derivative test.
  2. Test for local maximum/minimum at x = a if f(x) is not differentiable at x = a:
     
    Case 1: When f(x) is continuous at x = a and f′(ah) and f′(a + h) exist and are non-zero, then f(x) has a local maximum or minimum at x = a if f′(ah) and f′(a + h) are of opposite signs.
     
    If f′(ah) > 0 and f′(a + h) < 0 then x = a will be a point of local maximum.
     
    If f′(ah) < 0 and f′(a + h) > 0 then x = a will be a point of local minimum.
     
    Case 2: When f(x) is continuous and f′(ah) and f′(a + h) exist but one of them is zero, we should infer the information about the existence of local maxima/minima from the basic definition of local maxima/minima.
     
    Case 3: If f(x) is not continuous at x = a and f′(a h) and/or f′(a + h) are not finite, then compare the values of f(x) at the neighbouring points of x = a.
     
    Remark: It is advisable to draw the graph of the function in the vicinity of the point x = a, because the graph would give us the clear picture about the existence of local maxima/minima at x = a.

Concept of global maximum/minimum

Let y = f(x) be a given function with domain D. Let [a, b] D. Global maximum/minimum of f(x) in [a, b] is basically the greatest/least value of f(x) in [a, b]. Global maximum and minimum in [a, b] would occur at critical point of f(x) within [a, b] or at the end points of the interval.

Global maximum/minimum in [a, b]

In order to find the global maximum and minimum of f(x) in [a, b], find out all the critical points of f(x) in (a, b). Let c1, c2, …, cn be the different critical points. Find the value of the function at these critical points. Let f(c1), f(c2), …, f(cn) be the values of the function at critical points.
 
Say, M1 = max{f(a), f(c1), f(c2),…, f(cn), f(b)} and M2 = min{f(a), f(c1), f(c2), …, f(cn), f(b)}. Then M1 is the greatest value of f(x) in [a, b] and M2 is the least value of f(x) in [a, b]

Global maximum/minimum in (a, b)

Method for obtaining the greatest and least values of f(x) in (a, b) is almost same as the method used for obtaining the greatest and least values in [a, b], however with a caution.
 
Let y = f(x) be a function and c1, c2, …, cn be the different critical points of the function in (a, b).
 
Let M1 = max. {f(c1), f(c2), f(c3), …, f(cn)} and M2 = min.{f(c1), f(c2), f(c3), …, f(cn)}. Now if 88639.png or < M2, f(x) would have not global maximum (or global minimum ) in (a, b) .
 
This means that if the limiting values at the end points are greater than M1 or less than M2, then f(x) would not have global maximum/minimum in (a, b). On the other hand if M1 > 88633.png and M2 < 88627.png, then M1 and M2 would respectively be the global maximum and global minimum of f(x) in (a, b).

 

Notes:
  • For a continuous function maximum and minimum value occurs alternately.
  • If a function is discontinuous at a point xa it may have maximum value although it decreases on the left and increases on the right side of x = a.




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