Step 1: Find the 1 ^{st} Derivative, ${f}^{\prime}\left(x\right)$.
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 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 .
Most people remember the first option, the derivative equals zero, but most people forget about the second option, where the derivative does not exist .
Also, critical values are the places where our original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$, has either a horizontal tangent lines (zero slope), or is not differentiable (so not smooth or not continuous). These are the possible locations of maxes and mins; they are not always a max or a min.
Step 3: Do the actual 1 ^{st} Derivative Test (number line game).
This is the point in the method where you actually do the 1 ^{st} Derivative Test.
Remember that the endpoints of an x-interval are also critical values and should be marked as the endpoints of your number line.
If you have endpoints on your number line due to a closed x-interval , you do not need to mark test values outside the closed x-interval .
Plug your test values into your 1 ^{st} Derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$. Remember math people are not creative namers. It is called the 1 ^{st} Derivative Test because you are testing everything in the 1 ^{st} Derivative, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\prime}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.
Including the drawing of the directional arrow that goes with that behavior. Believe me a simple thing like drawing your directional arrows will really help you see what is happening. This gets back to our derivative triangle.
When doing a 1 ^{st} Derivative Test, we don’t care about the actual value of the derivative only if it is positive or negative.
Step 4: Draw Conclusions
Now that you have completed the 1 ^{st} Derivative Test, you read that 1 ^{st} Derivative information from the number line to answer actual question .
If your critical values do not have either of these shapes, that is fine, it means that it is neither a max or min, and was only a critical value . This is why you perform the 1 ^{st} Derivative Test; you must prove that your critical value has the behavior of a max or min.
To find the actual max or min y-value , you will need to plug your critical value , x-value , back into the original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$. The max or min is a point, ( x , y ) , on the original equation, $\textcolor[rgb]{}{f}\textcolor[rgb]{}{\left(}\textcolor[rgb]{}{x}\textcolor[rgb]{}{\right)}$.
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 usually not want to include the endpoints of your intervals because the endpoints will be your critical values , and at your critical values the function is not increasing or decreasing. When you need to connect multiple intervals for a solution you will need to use a union symbol , $\left(\textcolor[rgb]{}{a}\textcolor[rgb]{}{,}\textcolor[rgb]{}{b}\right)\mathit{\cup}\left(\textcolor[rgb]{}{c}\textcolor[rgb]{}{,}\textcolor[rgb]{}{d}\right)$.