Now understand the question. It is of the form,
IF Lab is the line with the steepest gradient then (%%%%%%%)
Where %%%%%%% is one of the options in the question.
This is what is meant by necessary. It means that for the statements of the form p->q, q has to be true if we have p as true.
Now the question asks which of the options has to be true (definitely) if the premise (i.e. Lab us the line with the steepest gradient) is true.
OPTION A:
Here adjacent points wrt to x co-ordinates mean, that there exists no point having an x co-ordinate b/w these two points.
Now based on this, let us see, option A. Suppose for contradiction, assume that option A is not true. Then, for two adjacent points with respect to x co-ordinates, shall not have the steepest slope, rather two nonadjacent points wrt to x co-ordinate will have the steepest slope.
The (Xp,Yp) is the intermediate point adjacent to (Xa,Xb). We supposed that Lab (the purple line) where Pa and Pb are non-adjacent wrt x co-ordinates is having the steepest gradient, but due to point (Xp.Yp), we got a steeper gradient, indicated by the orange line. So we reach a contradiction. So OPTION A is correct.
OPTION B: it is not a necessary condition for the y co-ordinates to be maximum and or minimum, because, the gradient is given by the ratio delta y/ delta x. So it shall also depend on the difference between the x coordinates.
OPTION C:
The difference b/w Xa and Xb need not be minimum, necessarily. It shall depend on the y co-ordinates as well. For counter example: