Show that the equation has a real root other than
Consider the function ( f(x) = x^3 + x^2 - 2x - 2 ). We are asked to show it has a real root other than ( x = -1 ). First, check if ( x = -1 ) is a root: [ f(-1) = (-1)^3 + ________ _____ ___ _____ ____ ____ _____.
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Using the principle of mathematical induction, show that
Show that the function defined by
has an inverse by applying the inverse function theorem. Find its inverse also.
Apply Bonnet Mean Value Theorem for integrals to show that
Determine the local minimum and local maximum values of the function f defined by .
Verify the second mean value theorem for the function and
in the interval
.
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.
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.