I read a question and its explanation, and immediately had a doubt. the question was: $y = ax^3 + bx^2 + cx + 5$ is a curve which goes through $A(-2,0)$..blah blah..
d question was to find values of a,b,c
in the explanation, they took out the derivative of equation:
$$y' = 3ax^2 + 2bx + c$$
and in d above equation, they substituted x by -2 & equated it to 0, so it became:
$$0 = 12a -4b + c$$
can this be done? it was as if they treated y' as a function in x separately and assumed that it too passed through the point $(-2,0)$
another way of looking at it is that they figured out that dy/dx at $x= -2$ will be 0. (But I don't think this is correct)
so my question is, if a curve passes through some point, will its derivative also pass through the same point?