Defining Local Extrema with the Second Derivative Test in Calculus

📚 Quick Study Guide

• 🔍 Definition of Local Extrema: A local maximum occurs at a point $c$ if $f(c) \geq f(x)$ for all $x$ near $c$. A local minimum occurs at a point $c$ if $f(c) \leq f(x)$ for all $x$ near $c$.
• 🔢 First Derivative Test: Find critical points where $f'(x) = 0$ or $f'(x)$ is undefined.
• 🧪 Second Derivative Test:
  • Calculate the second derivative, $f''(x)$.
  • Evaluate $f''(c)$ at each critical point $c$.
  • 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; use the first derivative test.
• 📝 Example Formula: If $f(x) = x^3 - 6x^2 + 5$, then $f'(x) = 3x^2 - 12x$ and $f''(x) = 6x - 12$.
• 💡 Important Note: The second derivative test only helps to identify local extrema; it doesn't find absolute extrema.

Practice Quiz

1. Which of the following is the correct statement of the Second Derivative Test?
   1. A) If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local maximum.
   2. B) If $f'(c) = 0$ and $f''(c) < 0$, then $f(c)$ is a local minimum.
   3. C) If $f'(c) = 0$ and $f''(c) > 0$, then $f(c)$ is a local minimum.
   4. D) If $f'(c) > 0$, then $f(c)$ is a local minimum.
2. Given $f(x) = x^3 - 3x$, what are the critical points?
   1. A) $x = 0, 1$
   2. B) $x = -1, 1$
   3. C) $x = 0, -1$
   4. D) $x = -3, 3$
3. For the function $f(x) = x^3 - 6x^2 + 5$, what is $f''(x)$?
   1. A) $3x^2 - 12x$
   2. B) $6x - 12$
   3. C) $6x + 12$
   4. D) $3x - 6$
4. If $f'(2) = 0$ and $f''(2) = -3$, what can you conclude about $f(2)$?
   1. A) $f(2)$ is a local minimum.
   2. B) $f(2)$ is a local maximum.
   3. C) $f(2)$ is an inflection point.
   4. D) The test is inconclusive.
5. Consider $f(x) = x^4$. What does the second derivative test tell us about the critical point at $x=0$?
   1. A) $f(0)$ is a local minimum.
   2. B) $f(0)$ is a local maximum.
   3. C) $f(0)$ is an inflection point.
   4. D) The test is inconclusive.
6. For $f(x) = x^2 - 4x + 7$, find the local minimum.
   1. A) $x = 0$
   2. B) $x = 1$
   3. C) $x = 2$
   4. D) $x = 3$
7. What happens if $f''(c) = 0$ in the second derivative test?
   1. A) $f(c)$ is a local minimum.
   2. B) $f(c)$ is a local maximum.
   3. C) $f(c)$ is an inflection point.
   4. D) The test is inconclusive.