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 real-valued 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
-
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.
- If
-
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 maxima -
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.
-
- Find values of x for which
-
Related Gate Questions:
- Gate CS 2012
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