When using the First Derivative Test, many students make a few common mistakes. I’ve learned from my own experiences, and I want to share some tips to help you avoid these errors.

1. Missing Critical Points

One big mistake is not finding all the critical points. Critical points happen when the derivative, noted as $f'(x)$, is either zero or doesn't exist. It’s important to solve $f'(x) = 0$ carefully and check for any points where the derivative is undefined. If you miss even one critical point, it can mess up your whole analysis.

2. Incorrect Sign Analysis

After finding the critical points, some students make mistakes when looking at the signs of the derivative in the different areas between those points. To use the First Derivative Test right, you should test points in each interval. For example, if your critical points are at $x = a$ and $x = b$, choose test values from the intervals $(-\infty, a)$, $(a, b)$, and $(b, \infty)$. If $f'(x)$ changes from positive to negative, you have a local maximum. If it goes from negative to positive, that means there is a local minimum.

3. Confusing Increasing and Decreasing

Another common mistake is misunderstanding what it means for a function to increase or decrease. Just because $f'(x) > 0$, it doesn't mean the function has a maximum. Even when the derivative changes signs, you need to look closely at the nearby intervals before jumping to conclusions.

4. Forgetting to State Results Clearly

Finally, it’s very important to explain your results clearly. Just saying that you’ve found a maximum or minimum isn’t enough. You need to mention the x-coordinate of the critical points and the value of the function at those points. For example, you should say something like, "At $x = c$, $f(x)$ has a local maximum of $f(c)$."

By remembering these common mistakes and using the First Derivative Test step by step, you’ll get a better understanding and see better results!