Might be I am asking a silly question but really want to clarify that would critical points and roots are same terms use interchangeably? I mean we can use critical point as value of x and root also as value of x then what is the difference between?
It has been my experience that a critical value is an input to a function such that the function's first derivative is either zero or undefined. A function's root(s) is/are the input(s) which cause the function to return zero, also known as the zeroes of a function.
One additional point- critical roots are numbers while critical points are, of course, points. If I were asked to find the critical root of $y= x^2- 6x+ 10= (x- 3)^2+ 1$, I would answer x= 3. If I were asked for the critical point, I would answer (3, 1).