The 2
^{nd}
Derivative Test method is almost identical to the 1
^{st}
Derivative Test method. Basically, take the 1
^{st}
Derivative on every step of the 1
^{st}
Derivative Test, and instead use the 2
^{nd}
Derivative. The meanings are different, but the steps are the same.

**
Step 1:
** Find the 2
^{nd}
Derivative, $f\prime \prime \left(x\right)$.

**
Step 2:
** Find the location of the possible inflection pointswhere $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=\textcolor[rgb]{}{0}$ or $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=\mathrm{}\textcolor[rgb]{}{\mathit{Does}}\textcolor[rgb]{}{\mathrm{}}\textcolor[rgb]{}{\mathit{Not}}\textcolor[rgb]{}{\mathrm{}}\textcolor[rgb]{}{\mathit{Exist}}\textcolor[rgb]{}{\mathrm{}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{\mathit{DNE}}\textcolor[rgb]{}{\right)}$

Inflection points can possibly occur where the 2
^{nd}
Derivative equals zero, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=0$, or does not exist, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=\mathrm{}\mathit{Does}\mathrm{}\mathit{Not}\mathrm{}\mathit{Exist}\mathrm{}\left(\mathit{DNE}\right)$.

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

–
To look for *
inflection points
*where$\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=\textcolor[rgb]{}{\mathit{Does\; Not\; Exist}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{\mathit{DNE}}\textcolor[rgb]{}{\right)}$, 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 second 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, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.

**
Step 3:
** Do the 2
^{nd}
Derivative Test (number line game)

– Draw a number line.

If you are provided an *
x-interval
*, [*
a
**
,
**
b
*], make those endpoints the ends of your number line.

–
Mark all of your *
possible
**
inflection points
*on the number line.

–
Choose test values on either side (left and right) of your *
possible
**
inflection points
*, and mark them on the number line.

–
Do the actual 2
^{nd}
Derivative Test.

Plug your test values into your 2
^{nd}
Derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$. Remember math people are not creative namers. It is called the 2
^{nd}
Derivative Test because you are testing everything in the 2
^{nd}
Derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.

– Mark the results either positive (concaveup) or negative (concave down) on your number line.

Include a drawing of the concavity that goes with that behavior.

**
Step 4:
** Draw Conclusions

Now that you have completed the 2
^{nd}
Derivative Test, you read that 2
^{nd}
Derivative information from the number line to answer *
actual question
*.

–
If you have been asked to determine the intervals, *
x-intervals
*, where the original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, is concaveup or concave down, they will generally want to see that answer in what is called interval notation (how convenient). They will generally want to see that answer in what is called *
interval notation
* (how convenient). *
Interval notation
* looks like (*
a
**
,
**
b
*) or [*
a
**
,
**
b
*]or some combo (*
a
**
,
**
b
*]. Where parenthesis, ( ), mean the *
endpoint
* is *
not included
*, and brackets, [ ], mean the *
endpoint
*is *
included
*. You will *
not want to include
* the *
endpoints
*at your *
endpoints
*the function is __
not
__concave up or concave down. When you need to connect multiple intervals for a solution you will need to use a *
union symbol
*, $\left(\textcolor[rgb]{}{a}\textcolor[rgb]{}{,}\textcolor[rgb]{}{b}\right)\mathit{\cup}\left(\textcolor[rgb]{}{c}\textcolor[rgb]{}{,}\textcolor[rgb]{}{d}\right)$.

–
If you have been asked to identify *
inflection points
*you will need to make sure the point fits both requirements of an *
inflection point
*.

1) $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=\textcolor[rgb]{}{0}$ or ${\textcolor[rgb]{}{f}}^{\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\prime}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}=\textcolor[rgb]{}{\mathit{DoesNotExist}}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{\mathit{DNE}}\textcolor[rgb]{}{\right)}$

2) The function *
must change
**
concavity
*at that *
x-value
*.

–
If your *
x-value
* does meet both of those requirements, then you will need to plug that *
x-value
* back into the original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, to find the *
y-value
*that goes with it. Remember at this stage in the process you only have the *
x-value
* of your *
inflection point
*, and that the *
y-value
* is required to create the actual (*
x
**
,
**
y
*) of the *
inflection point
*.

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