Step-by-Step Guide: Applying the Second Derivative Test for Extrema

📚 Quick Study Guide

🔍 Critical Points: Find where $f'(x) = 0$ or $f'(x)$ is undefined.
📈 Second Derivative: Compute $f''(x)$.
🧪 The Test:
• If $f''(c) > 0$, then $f(c)$ is a local minimum.
• If $f''(c) < 0$, then $f(c)$ is a local maximum.
• If $f''(c) = 0$, the test is inconclusive.

Practice Quiz

1. Question 1: What is the first step in applying the Second Derivative Test?
   1. Finding the second derivative.
   2. Finding the critical points.
   3. Setting the function equal to zero.
   4. Integrating the function.


2. Question 2: If $f''(c) > 0$ at a critical point $c$, what does this indicate?
   1. $f(c)$ is a local maximum.
   2. $f(c)$ is a local minimum.
   3. The test is inconclusive.
   4. $f(c)$ is an inflection point.


3. Question 3: If $f''(c) = 0$ at a critical point $c$, what does this mean for the Second Derivative Test?
   1. $f(c)$ is a local maximum.
   2. $f(c)$ is a local minimum.
   3. The test is inconclusive.
   4. $f(c)$ is an inflection point.


4. Question 4: Find the critical points of the function $f(x) = x^3 - 6x^2 + 5$.
   1. $x = 0, 2$
   2. $x = 0, 4$
   3. $x = 2, 4$
   4. $x = -2, 4$


5. Question 5: Given $f(x) = x^3 - 6x^2 + 5$, find $f''(x)$.
   1. $6x - 6$
   2. $3x^2 - 12x$
   3. $6x - 12$
   4. $2x - 6$


6. Question 6: Using the Second Derivative Test on $f(x) = x^3 - 6x^2 + 5$, what is the nature of the critical point at $x = 0$?
   1. Local Maximum
   2. Local Minimum
   3. Inflection Point
   4. Test Inconclusive


7. Question 7: Using the Second Derivative Test on $f(x) = x^3 - 6x^2 + 5$, what is the nature of the critical point at $x = 4$?
   1. Local Maximum
   2. Local Minimum
   3. Inflection Point
   4. Test Inconclusive