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Unit 9: Maxima and Minima: One Variable

Thus, at x = a, y = f(a) = BM Note
Now in small neighborhood at LHS of point B, take a point B for which x = a – h = OM , where h is
1
1
smallest.
Thus, at x = a – h, y = f(a-h) = B M > BM
1 1
Now in small neighborhood at LHS of point B, take a point B for which x = a + h = OM .
2 2
Thus, at x = a + h, y = f(a+h) = B M > BM
2 2
Now since B M is greater than BM, viz B M > BM
1 1 1 1
Thus, f(a-h) > f(a) or f(a) < f(a-h)
And since B M is greater than BM, viz B M > BM
2 2 2 2
Thus, f(a+h) > f(a) or f(a) < f(a+h)
Thus at point B for which x = a, value of f(x) viz the corresponding value of f(x) at the left or right side
of point B, f(a) viz f(a-h) and f(a+h) is smaller
At the point x = a, the function f(x) is called minimum if

f(a – h) > f(a) < f(a + h)
viz At x=a, the value f(a) of f(x) is smaller than both the value f(a-h) and f(a+h) in its small neighborhood.
The maximum value of any function does not mean that it is the biggest value and similarly
minimum value of does not mean that it’s the smallest value. There can be many maximum and
minimum value of any function and it is possible that a maximum value is smaller than minimum
value. At A, maximum value of function or degree is there, it only means that in the small
neighborhood of this point, its value is maximum and similarly in the small neighborhood of this
point, its value is minimum.

9.3 Conditions for Finding Maxima and Minima

Following are the conditions to find maximum and minimum of function y = f(x) at point x = a:
(i) Necessary condition – the essential condition for both maximum and minimum is as under:

dy
f ′′ ′′ ′ (x) = 0 or = 0
dx
(ii) Sufficient condition - the sufficient condition for both maximum and minimum is as under:
For maximum
2
dy
At x = a, the value of = negative value
dx 2
For minimum
2
dy
At x = a, the value of = 2 positive value
dx
Self Assessment

1. Fill in the blanks:
1. One of the main uses of mathematics is to determine the maxima and minima of any ………….
2. There can be many maximum and …………………… value of any function.

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