In a certain interval, it is possible to determine
the maximum and minimum values of the function when we know the
critical points of the function. The extreme value theorem guarantees
both the maximum and minimum value of a function under certain conditions.
It is stated as:
If a function f (x) is continuous on a closed interval
[a, b], then f (x) has both maximum and minimum value on [ a, b]
I) First establish the continuity of f (x)
on [a, b]
II) Determine all critical points of f (x) in
III) Evaluate the function f (x) at these critical
points and at the extremities of [a, b]
IV) The largest value is the maximum value and
the smallest value is the minimum value of f (x) on [a, b]
I) If f ' (x) > 0 for all x in the interval
I, then 'f ' is an increasing in I and
II) If f ' (x) < 0 for all x in
I then 'f ' is decreasing function in I. Is the converse of these
two rules true?
Justify your answer with a suitable example.
Solution : Consider the curve y = x3
as shown in the figure.
From the graph the curve is increasing at x = 0.
But f ' (0) =0 i.e. f ' (0) is not positive Hence the converse of
(1) is not true. In a similar way by considering y = x3
at x = 0, we can show that the curve is decreasing at x = 0 but
f ' (0) = 0. Hence the curve decreasing at x = 0 but f ' (0) is
not negative i.e. converse of (II) is also true.
Thus f ' ( c ) > 0 is only a sufficient condition
for f (x) to be increasing at x = c and f ' (c) < 0 is only a
sufficient condition for f (x) to be decreasing at x = c but it
is not a necessary condition.