second derivative test examples grade 12

📚 Quick Study Guide

• 📈 Critical Points: Find where $f'(x) = 0$ or is undefined.
• 🧪 Second Derivative: Compute $f''(x)$.
• 🔍 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. Use another method.
• 📝 Inflection Points: Points where the concavity changes. Find where $f''(x) = 0$ or is undefined. Check the sign of $f''(x)$ on either side of these points.

Practice Quiz

1. What does $f''(x) > 0$ imply at a critical point?
   1. Local maximum
   2. Local minimum
   3. Inflection point
2. If $f'(2) = 0$ and $f''(2) = -3$, what can you conclude about $f(x)$ at $x = 2$?
   1. Local minimum
   2. Local maximum
   3. Inflection point
3. The second derivative test is inconclusive when:
   1. $f''(x) > 0$
   2. $f''(x) < 0$
   3. $f''(x) = 0$
4. What is an inflection point?
   1. Where the function reaches its highest value
   2. Where the concavity of the function changes
   3. Where the function reaches its lowest value
5. If $f''(x) = 6x - 4$, at what x-value might there be an inflection point?
   1. $x = 2/3$
   2. $x = -2/3$
   3. $x = 3/2$
6. Given $f(x) = x^3 - 6x^2 + 5$, find $f''(x)$.
   1. $6x - 12$
   2. $3x^2 - 12x$
   3. $x^3 - 12x$
7. Using the second derivative test, determine the nature of the critical point at $x = 0$ for $f(x) = x^4$.
   1. Local minimum
   2. Local maximum
   3. Test is inconclusive