Method: 2nd Derivative Test

Step 1: Find the first derivative, $f\prime \left(x\right)$, and the second derivative, $f\prime \prime \left(x\right)$.

Step 2: Find the critical values.

• $f\prime \left(x\right)=0$
• $f\prime \left(x\right)= \mathit{Does\; Not\; Exist }\left(\mathit{DNE}\right)$

Step 3: Plug your critical values into your second derivative, $f\prime \prime \left(x\right)$.

Step 4: Draw conclusions based upon the Step 3 information.

• If $f\prime \prime \left(\mathit{critical\; value}\right)>0$ or positive, then you know that you have a local min at that critical value .
• If $f\prime \prime \left(\mathit{critical\; value}\right)<0$ or negative, then you know that you have a local max at that critical value .
• If $f\prime \prime \left(\mathit{critical\; value}\right)=0$, then you know that your test failed , and you don’t know anything about your final answer. It still could be a local max, local min, but all you know is that the test failed and you wasted a bunch of your time.