The method you will use to find global extrema is almost identical to the 1
^{st}
Derivative Test method. The nice part about global extrema method is that you will *
not
* have to apply the actual 1
^{st}
Derivative Test step (number line game).

**
Step 1:
** Find the 1
^{st}
Derivative, ${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.

**
Step 2:
** Find the *
critical values
*of the equation.

Critical values are the places where the derivative equals zero, ${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}\textcolor[rgb]{}{=}\textcolor[rgb]{}{0}$, or *
does not exist
*, ${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}\textcolor[rgb]{}{=}\textcolor[rgb]{}{\mathit{DNE}}$.

–
To look for *
critical values
*where ${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}\textcolor[rgb]{}{=}\textcolor[rgb]{}{0}$, take your derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, and set it equal to zero and solve for x.

–
To look for critical values where${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=\textcolor[rgb]{}{\mathit{Does\; Not\; Exist}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{\mathit{DNE}}\textcolor[rgb]{}{\right)}$, you want to look for places that would “break math”. These “breaking math” locations are most often going to occur when you have a *
fraction
*, and you have to consider when you would get *
division
* by zero. *
Division
* by zero is an example of “breaking math”, you are not allowed to *
divide
* by zero. Whatever *
x-value
*would cause that to happen would “break math”.

To help yourself identify fractions in your derivative, look for *
negative exponents
*. If you see a *
negative exponent
* you will want to apply your Laws of Exponents rewrite in order to turn all your *
negative exponents
**
positive
*. I always say *
negative exponents
* are *
great
* for doing *
calculus
* they *
suck
* for doing *
algebra
*.

–
Other examples of “breaking math” would be *
square roots
* of *
negative numbers
*, *
logarithms
* of *
negative numbers
*, or *
x-values
*that were not in the *
domain
*of your original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.

–
If you have a closed interval, [*
a
**
,
**
b
*], then the endpoints of that interval would also be places where, ${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}\textcolor[rgb]{}{=}\textcolor[rgb]{}{\mathit{DNE}}$, and so you would want to consider them as *
critical values
*as well.

*
REMEMBER
*: Check that all of your

**
Step 3:
** Plug the *
x-values
*of your *
critical values
* and your endpoints of your closed interval, [*
a
**
,
**
b
*], back into your original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$. This will give you the *
y-values
*that go with these *
x-values
*.

**
Step 4:
** Draw conclusions.

–
The largest *
y-value
*, is the winner of the Absolute/Global Max award.

The answer would be given like this: “The absolute/global max is *
y-value
*, and it occurs at these

Remember you could have an Absolute/Global max (*
y-value
*) that occurs at multiple locations (*
x-value
*).

–
The smallest *
y-value
*, is the winner of the Absolute/Global Min award.

The answer would be given like this: “The absolute/global min is *
y-value
*, and it occurs at these

Again, you could have an Absolute/Global min (*
y-value
*) that occurs at multiple locations (*
x-value
*).

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