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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Apply Bonnet Mean Value Theorem for integrals to show that
a) If the partition P2 is a refinement of the partition P1 of [a,b], then and
. Verify this result for the function
defined over the interval
and the partitions
and
.
b) Evaluate: .
Write the inequality in the modulus form.
Show that is an algebraic number.
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.
Show that the equation has a real root other than
Using Weiestrass M-test, show that the following series converges uniformly.