Using the First Derivative Test to Determine Relative (Local) Extrema

Using the First Derivative Test to Determine Relative (Local) Extrema - AP Calculus AB/BC Study Guide

Hello, mathletes! 📊 Time to dive into the fabulous world of derivatives and discover how they can help you find those coveted relative extrema points in functions. Think of derivatives as your mathematical detective tools, helping you find where functions reach their highs and lows. Ready to flex those brain muscles? 💪

So, can the derivative of a function do more than just tell us if a function is going up, up, up, or down, down, down? Absolutely! It’s like having a superpower that lets you pinpoint those sweet spots called relative (or local) extrema. These are the local maximums and minimums, not the rollercoaster of your feelings when you see a calculus exam!

Hold your horses for a second. Make sure you’re up-to-date with key concepts such as the Extreme Value Theorem, the difference between global and local extrema, and those nifty critical points that make all this possible. If not, go back and review these topics in AP Calculus 5.2. Cool? Cool.

We can use the First Derivative Test to find out where a function hits its relative maximums and minimums. Here’s the scoop:

1. When the derivative of a function changes from positive to negative at a point, it means the function switches from increasing to decreasing. Voilà! You’ve got yourself a local maximum! 🎉

2. When the derivative changes from negative to positive, it means the function is like, "Hey, I'm done decreasing; let’s start increasing!" This means you’ve found a local minimum. 🥂

The process goes like this: first, spot the critical points—places where the derivative is zero or doesn't exist. Then, put on your detective hat and sniff out the signs of the derivative on either side of these points. Is it positive on the left and negative on the right? Boom, local max! Negative on the left and positive on the right? Bam, local min!

Let’s roll through an example.

Consider the function ( f(x) = x^2 ). We know (thanks to the power rule) that the derivative is ( f′(x) = 2x ). Spoiler alert: the derivative at ( x = 0 ) is zero, so we have a critical point.

Checking the Left Side (for ( x < 0 )): Imagine us being Sherlock for a second. Try ( x = -1 ):

[ f′(-1) = 2(-1) = -2 ]

The derivative is negative, making the function decrease before the critical point.

Checking the Right Side (for ( x > 0 )): Now, let's plug in ( x = 1 ):

[ f′(1) = 2(1) = 2 ]

The derivative is positive, making the function increase after the critical point.

Conclusion: The derivative changes from negative to positive, so ( x = 0 ) is a local minimum. Hoo-rah! 🎉⬇️

📝 First Derivative Test Practice Problems

Your turn, math wizard! Try these out:

Problem 1: Let ( h(x) ) be a polynomial function with ( h′(x) = x^2(x - 3)(x + 4) ). Where are ( h )'s relative minima?

Problem 2: Consider ( g(x) = x^5 - 80x ). Where are ( g )'s relative maxima?

✅ Solutions

Solution to Problem 1:

Find the critical points first:

[ x^2 = 0 \longrightarrow x = 0 ] [ (x - 3) = 0 \longrightarrow x = 3 ] [ (x + 4) = 0 \longrightarrow x = -4 ]

Let's check the sign of the derivative around each critical point:

1. At ( x = -4 ):

   • Left side: Positive
   • Right side: Negative
   • Verdict: Relative maximum

2. At ( x = 0 ):

   • Left side: Negative
   • Right side: Negative
   • Verdict: Not an extremum

3. At ( x = 3 ):

   • Left side: Negative
   • Right side: Positive
   • Verdict: Relative minimum

So, ( h(x) ) has a relative minimum at ( x = 3 ).

Solution to Problem 2: Find the critical points which occur where ( g′(x)) is zero:

[ g′(x) = 5x^4 - 80 ] [ 5x^4 - 80 = 0 ] [ x = \sqrt[4]{16} = \pm 2 ]

1. At ( x = -2 ):

   • Left side: Positive
   • Right side: Negative
   • Verdict: Relative maximum

2. At ( x = 2 ):

   • Left side: Negative
   • Right side: Positive
   • Verdict: Relative minimum

So, ( g(x) ) has one relative maximum at ( x = -2 ).

🌟 Wrapping It Up

Fantastic job, brainy bunch! You just unlocked the superpower of using the First Derivative Test to detect relative extrema. By identifying those critical points and analyzing the signs of the derivative, you can pinpoint where a function hits its highs and lows within specific intervals. Now take a moment to bask in the glory of your newfound knowledge, and as always, practice makes perfect!

Key Terms to Review

1. Decreasing Function: Values go down as you move right on the graph.
2. Derivative: Measures how a function changes.
3. First Derivative Test: Determines increasing or decreasing intervals and identifies local extrema.
4. Global Extremum: The highest (maximum) or lowest (minimum) point over the entire graph.
5. Increasing Function: Values go up as you move right on the graph.
6. Relative Maximum: The highest point within a specific interval.
7. Relative Minimum: The lowest point within a specific interval.
8. Second Derivative Test: Analyzes concavity to determine maxima and minima.

Now go forth, crush those problems, and may the derivatives be ever in your favor! 🚀📈