Consider the function $\displaystyle f:R\rightarrow R$
$\displaystyle
f(x)=(x-1)(x-2)^2(x-3)^3(x-4)^4(x-5)^5......(x-n)^n
$
Find the number of points at which the function has an extremum(i.e points of local maxima/minima).
considering the fact that the graph of a polynomial at the roots with even (odd) multiplicity touches (crosses) the x-axis, and hoping that nothing weird happens in between, i believe that the
number of required points is: $\displaystyle n + \left \lfloor \frac{n}{2} \right \rfloor -1.$