# What are the values and types of the critical points, if any, of #f(x)=x / (x^2 + 4)#?

##### 2 Answers

#### Explanation:

Let

Denominator goes to zero when we multiply to the other side to solve for x.

Since the denominator will never equal to zero we only have one type of critical points and that is stationary

The critical points are

#### Explanation:

There are several different uses of "critical point" and "types of critical point".

**Meaning of "critical point"**

In the terminology I was taught (and that I teach), a critical point of a function

With that definition, we proceed:

**"types" of critical points**

Again, using the terminology I am familiar with, a critical point may be the location of a local minimum, a local maximum or neither. (In addition to "local" I also use "relative" -- they mean the same thing.)

Applying the first derivative text we find that there is a local minimum at

That minimum is

Applying the first derivative text we find that there is a local maximum at

That minimum is

**Other uses of the words**

I have seen some who define a critical point as a point on the graph of a function where the derivative is

The would say that the critical points are

Some would also say that