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Critical points and Roots?


New member
May 3, 2020
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?


Staff member
Feb 24, 2012
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.


Well-known member
MHB Math Helper
Mar 1, 2012
In solving rational inequalities, critical values are where the expression under scrutiny equals zero or is undefined. Those values mark the boundaries for the solution set.

for example ...

$\dfrac{x^2-1}{x+2} \ge 0$

critical values are $x \in \{-2,-1,1\}$

the three critical values partition the set of x-values into four regions ...

$-\infty < x < -2$,
$-2 < x < -1$,
$-1 < x < 1$,
and $x > 1$

The "equals to" part of the original inequality occurs at $x = \pm 1$

The "greater than" occurs over the intervals $-2 < x < -1$ and $x > 1$

So, the solution set is all $x$ such that $-2 < x \le -1$ or $x \ge 1$

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
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).