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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a) Determine the points of discontinuity of the function f and the nature of discontinuity at each of those points:
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Also check whether the function f is derivable at x = 1.
Show that is an algebraic number.
Write the inequality in the modulus form.
Determine the local minimum and local maximum values of the function f defined by .
Check whether the intervals ]5,9] and [6,12[ are equivalent or not.
Using the principle of mathematical induction, show that
Verify the second mean value theorem for the function and
in the interval
.