#First Derivative Test

#Table of Contents

1. Introduction
2. Relationship Between First Derivative and Local Extrema
3. First Derivative Test
4. Worked Example
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

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#Introduction

Understanding how the first derivative of a function is related to its local extrema (maximums and minimums) is crucial in calculus. This study note will explore the first derivative test, a method used to identify and classify these extrema.

#Relationship Between First Derivative and Local Extrema

#Explanation

Local extrema are found at critical points where the first derivative, $f'(x)$, is equal to zero. However, not all points where $f'(x) = 0$ are local extrema. For example, the graph of $y = x^3$ has a derivative of zero at $x = 0$, but this point is not a minimum or maximum; it is a point of inflection.

A more precise definition for local minimums and maximums is:

• If $x = a$ is a critical point of $f(x)$ (i.e., $f'(a) = 0$) and if:
  • $f'(x)$ changes from positive to negative at $x = a$, then $x = a$ is a local maximum.
  • $f'(x)$ changes from negative to positive at $x = a$, then $x = a$ is a local minimum.
  • If the sign of $f'(x)$ does not change, $x = a$ is a point of inflection.

#First Derivative Test

#Steps

1. Find the Critical Points: Solve $f'(x) = 0$ to identify critical points.
2. Evaluate the First Derivative Around Critical Points:
   • Check the value of $f'(x)$ slightly to the left of the critical point.
   • Check the value of $f'(x)$ slightly to the right of the critical point.
3. Classify the Critical Points:
   • If $f'(x)$ changes from positive to negative, it is a local maximum.
   • If $f'(x)$ changes from negative to positive, it is a local minimum.
   • If $f'(x)$ does not change sign, it is a point of inflection.

#Summary Table

#Worked Example

#Problem

Find the coordinates of the critical points on the graph of $f(x) = 2x^3 + 3x^2 - 12x + 1$, and classify the nature of each point using the first derivative test.

#Solution

1. Find the Derivative: $$f'(x) = 6x^2 + 6x - 12$$

2. Find the Critical Points: $$\begin{aligned} 6x^2 + 6x - 12 &= 0 \\ x^2 + x - 2 &= 0 \\ (x + 2)(x - 1) &= 0 \\ \end{aligned}$$ Thus, $x = -2$ and $x = 1$.

3. Evaluate $f(x)$ at Critical Points: $$\begin{aligned} f(-2) &= 2(-2)^3 + 3(-2)^2 - 12(-2) + 1 = 21 \\ f(1) &= 2(1)^3 + 3(1)^2 - 12(1) + 1 = -6 \\ \end{aligned}$$ Critical points are $(-2, 21)$ and $(1, -6)$.

4. Classify the Points:

   • At $x = -2$: $$\begin{aligned} f'(-2.1) &> 0 \quad (\text{positive}) \\ f'(-1.9) &< 0 \quad (\text{negative}) \\ \end{aligned}$$ Changes from positive to negative, so $(-2, 21)$ is a local maximum.

   • At $x = 1$: $$\begin{aligned} f'(0.9) &< 0 \quad (\text{negative}) \\ f'(1.1) &> 0 \quad (\text{positive}) \\ \end{aligned}$$ Changes from negative to positive, so $(1, -6)$ is a local minimum.

#Conclusion

• Local maximum at $(-2, 21)$.
• Local minimum at $(1, -6)$.

#Practice Questions

Practice Question

1. Find the critical points and classify them for the function $f(x) = x^3 - 3x^2 + 4$.
2. Given $f(x) = x^4 - 4x^3 + 6x^2$, use the first derivative test to find and classify all local extrema.
3. For $f(x) = e^x - x^2$, determine the critical points and their nature.

#Glossary

1. Local Extrema: Points where a function reaches a local maximum or minimum.
2. Critical Points: Points where the derivative of a function is zero or undefined.
3. First Derivative Test: A method to determine whether a critical point is a local maximum, minimum, or point of inflection.

#Summary and Key Takeaways

#Summary

• The first derivative test is used to find and classify local extrema by analyzing the sign changes of the first derivative around critical points.
• Critical points occur where the first derivative equals zero or is undefined.
• The test involves checking the derivative values slightly to the left and right of the critical points.

#Key Takeaways

• Local maxima occur where $f'(x)$ changes from positive to negative.
• Local minima occur where $f'(x)$ changes from negative to positive.
• Points of inflection occur where $f'(x)$ does not change sign.

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By mastering the first derivative test, you can efficiently identify and classify local extrema, a vital skill in calculus and many applications.