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 domain of ( f ) _____ ________ ______ ____ ______.
_____ _________ ___ ______ ____ ___.
_____ _____ _________ ____ ___ ____ ______ ____ ____ ___.
_________ ________ ______ ___ _______ _______ _____ _____ ___ ____.
_______ _______ _____ ________ __________ __________ ______ __________.
_____ _________ ____ ____ ________.
___ _____ _________ _________ ________ ___ __________ __________.
_______ ___ ________ _____ ______ ________ _____.
_____ _________ ________ ____ ____ __________ ____ ___ ________ ______ _____ _________.
________ __________ _________ _________ _________ _______ __________.
________ ________ _____ __________ ______ ______ ____ _________ _________ _______ ___.
__________ __________ ___ _____ ___ _______ __________.
_____ ___ ____ __________ _____ ___ ____ _______ ______ _____ ___.
_____ _____ _____ _____ __________ ___ __________ ____ ___ _______ _______.
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Show that , the lagrange's form of remainder in the Maclaurin series expansion of e4x, tends to zero as
. Hence obtain the Maclaurin's infinite expansion for e4x.
Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?
a) Test the following series for convergence,
(i)
(ii)
b) Show that is conditionally convergent.
Use the Fundamental Theorem of Integral Calculus to evaluate the integral
Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.
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 .