Worked Problems: Second Derivative Test for Extrema with Full Solutions

📚 Quick Study Guide

🔍 The Second Derivative Test helps determine if a critical point of a function is a local maximum or local minimum. 💡 First, find the critical points of the function $f(x)$ by setting $f'(x) = 0$ and solving for $x$. 📝 Let $c$ be a critical point of $f(x)$. ➕ If $f''(c) > 0$, then $f(x)$ has a local minimum at $x = c$. ➖ If $f''(c) < 0$, then $f(x)$ has a local maximum at $x = c$. ❓ If $f''(c) = 0$, the test is inconclusive. Other methods must be used to determine the nature of the critical point. ➗ Remember to calculate the second derivative $f''(x)$ correctly.

Practice Quiz

1. What does a positive second derivative at a critical point indicate?
   1. Local maximum
   2. Local minimum
   3. Inflection point
   4. Saddle point
2. Find the critical points of $f(x) = x^3 - 6x^2 + 5$?
   1. $x=0, x=2$
   2. $x=0, x=4$
   3. $x=2, x=4$
   4. $x=-2, x=4$
3. Given $f(x) = x^4 - 4x^3 + 6$, find $f''(x)$.
   1. $12x^2 - 24x$
   2. $4x^3 - 12x^2$
   3. $4x^2 - 12x$
   4. $12x - 24$
4. 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
   4. The test is inconclusive
5. For $f(x) = x^3$, what does the second derivative test tell us about the critical point at $x=0$?
   1. Local minimum
   2. Local maximum
   3. Inflection point
   4. The test is inconclusive
6. Find the $x$-value of any inflection points for the function $f(x) = x^3 - 3x^2 + x - 1$.
   1. $x = 0$
   2. $x = 1$
   3. $x = -1$
   4. No inflection points exist.
7. A function has a critical point at $x = 1$. Given that $f''(1) = 5$, does the function have a local minimum or maximum at that point?
   1. Local minimum
   2. Local maximum
   3. Neither
   4. Cannot be determined