Are the following statements true or false? Give reasons for your answer.
a) Complement of the open interval ]0, 1] is an open set.
b) Every bounded sequences is not convergent.
c) The function defined by
is uniformly continuous.
d) If the first derivative of a function at a point vanishes, then it has an extreme value at that point.
e) The function defined by
is not integrable.
a) False. The complement of the interval ]0, 1] in ℝ is (−∞, 0] ∪ (1, ∞), which contains closed endpoints 0 and 1. Since it includes its boundary points, it is not ________ ________ _____ ______ ______ ____ __________.
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Show that is an algebraic number.
Using the principle of mathematical induction, show that
Apply Bonnet Mean Value Theorem for integrals to show that
Determine the local minimum and local maximum values of the function f defined by .
Use Cauchy's mean value theorem to prove that:
Find a and b such that exists.
Verify Bozano-Weierstrass Theorem for the following sets:
i) Set of non-negative integers.
ii) Interval
d) Check whether the limit exists or not?