Prove that a strictly decreasing function is always one-one.
Let ( f ) be a strictly decreasing function. By definition, for any two points ( x_1 ) and ( x_2 ) in the _________ ____ __________ ____ __________ _________ __________.
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Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.
a) If the partition P2 is a refinement of the partition P1 of [a,b], then and
. Verify this result for the function
defined over the interval
and the partitions
and
.
b) Evaluate: .
a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
Use Cauchy's mean value theorem to prove that:
Using the principle of mathematical induction, show that
Using Weiestrass M-test, show that the following series converges uniformly.
Determine the local minimum and local maximum values of the function f defined by .