Let , where
has a positive root. Then,
has a root
such that
has at least one real root
has at least two real roots
All of the above
It is given that is a positive root of
and by inspection, we have
. Therefore,
are the roots of
.
By Rolle's theorem, has root
between 0 and
So option (a) is correct.
Clearly, is a fourth degree equation in x and imaginary roots always occur in pairs. Since,
is a root of
.
Therefore, will have another real root,
(say)
Now, are real roots of
.
Therefore, by Rolle's theorem will have a real root between
Thus, option (b) is correct.
We have see that are two real roots of
. As
is a fifth degree equation, it will have at least three real roots. Consequently, by Rolle's theorem
will have at least two real roots.
Thus, option (c) is also correct.
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