Let $$f(x)=(x-a)(x-b)^3(x-c)^5(x-d)^7 $$
are real numbers with
$a < b < c < d$
. Thus $ f
)$ has $16$ real roots counting multiplicities and among them $4$ are
distinct from each other. Consider
)$, i.e. the derivative of
)$. 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.