Let f(x)=(x−a)(x−b)3(x−c)5(x−d)7
where a,b,c,d
are real numbers with
a<b<c<d
. Thus f(x) has 16 real roots counting multiplicities and among them 4 are
distinct from each other. Consider
f′(x), i.e. the derivative of
f(x). Find the following:
(i) the number of real roots of
f′(x), counting multiplicities,
(ii) the number of
distinct
real roots of
f′(x).
This is a polynomial of degree 16 hence the derivative will be of degree 15 and hence it will have 15 roots. But are they real ?
How to find distinct real roots ? Rolle's theorem tells only about existence of root.
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