Troubleshooting: Why your First Derivative Test results are wrong

📚 Quick Study Guide

🔍 The First Derivative Test helps identify local maxima and minima of a function. 📈 Find critical points by setting the first derivative, $f'(x)$, equal to zero or identifying where it is undefined. ➕ Examine the sign of $f'(x)$ to the left and right of each critical point. 💡 If $f'(x)$ changes from positive to negative at $x=c$, then $f(c)$ is a local maximum. ➖ If $f'(x)$ changes from negative to positive at $x=c$, then $f(c)$ is a local minimum. ↔️ If $f'(x)$ does not change sign at $x=c$, then $f(c)$ is neither a local maximum nor a local minimum (it's a saddle point). ⚠️ Common errors include incorrect differentiation, algebraic mistakes when solving for critical points, and misinterpreting the sign changes of $f'(x)$.

Practice Quiz

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. How do you find critical points for the First Derivative Test?
1. (A) Set the original function, $f(x)$, equal to zero.
2. (B) Set the second derivative, $f''(x)$, equal to zero.
3. (C) Set the first derivative, $f'(x)$, equal to zero or find where it is undefined.
4. (D) Find the y-intercept of $f(x)$.
3. If $f'(x)$ changes from negative to positive at $x=c$, what does this indicate about $f(c)$?
1. (A) $f(c)$ is a local maximum.
2. (B) $f(c)$ is a local minimum.
3. (C) $f(c)$ is an inflection point.
4. (D) $f(c)$ is neither a maximum nor a minimum.
4. What does it mean if $f'(x)$ does not change sign at $x=c$?
1. (A) $f(c)$ is a local maximum.
2. (B) $f(c)$ is a local minimum.
3. (C) $f(c)$ is a point of discontinuity.
4. (D) $f(c)$ is neither a local maximum nor a local minimum.
5. Which of the following is a common error when using the First Derivative Test?
1. (A) Correctly differentiating the function.
2. (B) Accurately solving for critical points.
3. (C) Misinterpreting the sign changes of $f'(x)$.
4. (D) Understanding the domain of $f(x)$.
6. Suppose $f'(x) = 2x - 4$. What is the critical point of $f(x)$?
1. (A) $x = -2$
2. (B) $x = 0$
3. (C) $x = 2$
4. (D) $x = 4$
7. If $f'(x) = x^2$, what can you conclude about critical points and local extrema?
1. (A) There is a local maximum at $x=0$.
2. (B) There is a local minimum at $x=0$.
3. (C) There is a saddle point at $x=0$.
4. (D) There are no critical points.