Second Derivative Test Fails: Understanding and Overcoming Limitations

📚 Quick Study Guide

• 🔍 The second derivative test helps determine if a critical point (where $f'(x) = 0$ or is undefined) is a local maximum, local minimum, or saddle point.
• 📈 If $f''(c) > 0$ at a critical point $c$, then $f(c)$ is a local minimum.
• 📉 If $f''(c) < 0$ at a critical point $c$, then $f(c)$ is a local maximum.
• 🤷‍♀️ If $f''(c) = 0$ or $f''(c)$ is undefined, the second derivative test is inconclusive. Other methods are needed to determine the nature of the critical point.
• ✏️ When the second derivative test fails, you can use the first derivative test (analyzing the sign of $f'(x)$ around the critical point) or other techniques.
• 📏 Functions like $f(x) = x^4$ at $x=0$ demonstrate a case where $f'(0) = 0$ and $f''(0) = 0$, but $x=0$ is a local minimum.
• 🧪 Functions like $f(x) = -x^4$ at $x=0$ demonstrate a case where $f'(0) = 0$ and $f''(0) = 0$, but $x=0$ is a local maximum.
• ➗ Functions like $f(x) = x^3$ at $x=0$ demonstrate a case where $f'(0) = 0$ and $f''(0) = 0$, but $x=0$ is neither a local max nor a local min.

Practice Quiz

1. If $f'(c) = 0$ and $f''(c) = 0$, what does the second derivative test tell us about the point $c$?

   1. It is a local maximum.
   2. It is a local minimum.
   3. The test is inconclusive.
   4. It is an inflection point.

2. For the function $f(x) = x^4$, what happens at $x = 0$?

   1. It's a local maximum.
   2. It's a local minimum.
   3. It's neither a local max nor a local min.
   4. The second derivative test fails, but it's a saddle point.

3. Which test can be used when the second derivative test fails?

   1. The Integral Test
   2. The First Derivative Test
   3. The Ratio Test
   4. The Root Test

4. Consider $f(x) = -x^4$. What is the nature of the critical point at $x = 0$?

   1. Local minimum
   2. Local maximum
   3. Neither local max nor min
   4. Inflection Point

5. What is a critical point?

   1. A point where $f(x) = 0$
   2. A point where $f'(x) = 0$ or is undefined
   3. A point where $f''(x) = 0$
   4. A point where the graph crosses the x-axis

6. For what function does the second derivative test fail at $x = 0$, and it's neither a local maximum nor a local minimum?

   1. $f(x) = x^4$
   2. $f(x) = -x^4$
   3. $f(x) = x^3$
   4. $f(x) = x^2$

7. What does it mean for a test to be 'inconclusive'?

   1. The test proves the point is a maximum.
   2. The test proves the point is a minimum.
   3. The test does not provide enough information to determine the nature of the point.
   4. The test proves the point is an inflection point.