Wednesday, 16 August 2017

calculus - Number of real roots of f(x)

Let f(x)=(xa)(xb)3(xc)5(xd)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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