MHB Are Critical Points and Roots Interchangeable in Math?

[gevni]

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?

 

[MarkFL]

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.

 

[skeeter]

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 $x \ge 1$

 

[HOI]

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).