Difference between first derivative test and second derivative test

📚 Quick Study Guide

• 📈 The First Derivative Test helps find local maxima and minima of a function by analyzing the sign changes of the first derivative.
• 📉 A critical point $c$ is a local maximum if $f'(x)$ changes from positive to negative at $x=c$.
• ℹ️ A critical point $c$ is a local minimum if $f'(x)$ changes from negative to positive at $x=c$.
• ↔️ If $f'(x)$ does not change sign at $x=c$, then $c$ is neither a local maximum nor a local minimum (a saddle point).
• 🍎 The Second Derivative Test uses the second derivative to determine the concavity of the function at a critical point.
• 😀 If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local minimum.
• 😭 If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is a local maximum.
• 🤷 If $f'(c) = 0$ and $f''(c) = 0$, the test is inconclusive. Use the first derivative test.

Practice Quiz

1. Question 1: What does the first derivative test primarily help to identify?
   1. A) Inflection points
   2. B) Local maxima and minima
   3. C) Concavity
   4. D) Asymptotes
2. Question 2: If $f'(x)$ changes from negative to positive at $x=c$, what can be concluded about the point $c$?
   1. A) It is a local maximum
   2. B) It is a local minimum
   3. C) It is an inflection point
   4. D) It is a saddle point
3. Question 3: What does the second derivative test primarily help to determine?
   1. A) Increasing/Decreasing intervals
   2. B) Local extrema using concavity
   3. C) Roots of the function
   4. D) Asymptotes
4. Question 4: If $f'(c) = 0$ and $f''(c) > 0$, what can be concluded about $f(c)$?
   1. A) It is a local maximum
   2. B) It is a local minimum
   3. C) It is an inflection point
   4. D) The test is inconclusive
5. Question 5: If $f'(c) = 0$ and $f''(c) = 0$, what should you do?
   1. A) Conclude that it is an inflection point
   2. B) Use the second derivative test again
   3. C) Use the first derivative test
   4. D) Conclude that it is neither a maximum nor a minimum
6. Question 6: Which test is more suitable for identifying saddle points?
   1. A) First Derivative Test
   2. B) Second Derivative Test
   3. C) Both tests are equally suitable
   4. D) Neither test is suitable
7. Question 7: What does $f''(x)$ represent?
   1. A) The rate of change of the slope of the tangent line
   2. B) The slope of the tangent line
   3. C) The roots of the function
   4. D) The area under the curve