I actually don't know which chapter this question belongs to, can someone please change it for me.
4 Colinear points means four real roots
Equating second derivative to 0 means we are interested in finding the points of inflection of the curve
is the ans 6 ? @Soumyadeep
Yes it Is 6
So equating second derivative to 0 and saying it has real roots means it has two points of inflection but it may be that curve never crosser x axis still.
So is it because of this they have given that it has 4 collinear points and not 4 roots because 4 collinear point just means and line may cut it at 4 points not just the x axis?
@Soumyadeep is this gmp ?
If f(x) has 4 real roots then f'(x) has 3 real roots and f''(x) has 2 real roots. This theory is given in aod check any book @Soumyadeep
just a slight clarification. f(x) has four collinear points means that f(x) = Ax+B for some A,B
which means that f(x) - Ax -B=0 has four distinct roots. The rest of the solution proceeds as has been done in the other posts.