Determine the local minimum and local maximum values of the function f defined by
To find the local minimum and maximum values of the function \( f(x) = 3 - 5x^3 + 5x^4 - x^5 \), we need to follow these steps: ___ _________ _______ ______ ___ ______.
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Give an example for each of the following.
i) A set in with a unique limit point.
ii) A set in whose all points except the one are its limit points.
iii) A set having no limit point.
iv) A set S with
v) A bijection from to
Find the following limit
Suppose that is continuous on
and differentiable on
and that f )0( = ,0 f )1( = ,1 f )2( = .1 (i)
Show that there exists such that
Show that there exists such that
(iii) Show that there exists such that .
Test the following series for convergence.
(i)
(ii)
The product of two divergent sequences is divergent. True or false? Justify.
Let be any sequence. Show that
iff for every ε > 0, there exists some
such that n ≥ N implies
Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.