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Local Maximum and Minimum

Learn to identify, calculate, and apply local maxima and minima in functions through graph analysis and algebraic methods with derivative usage. Practice solving critical points to pinpoint these extrema effectively.

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Local Maximum and Minimum

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  1. Local Maximum and Minimum

  2. Objectives • Students will learn how to find the local min/max through the use of a graph and algebraically (with a number line)

  3. Prerequisite Concepts • Distinguish between absolute maximum and minimum • Understand where to find absolute maximum and minimum on a graph • Use a derivative to find critical points

  4. Example 1 Absolute max: occurs at x=2 Absolute min: occurs at x=-3

  5. Example 2 Find the critical numbers (where ) Find the derivative Solve for x Factor and set equal to zero

  6. Let’s Get Started!

  7. Local Maximum and Minimum • The smallest and largest y-values within a smaller interval on a function’s domain

  8. How to Find a Local Maximum and Minimum on a Graph: Local max: (slope changes from positive to negative) Local min: (slope changes from negative to positive)

  9. How to Find a Local Maximum and Minimum Using an Equation: • Derive • Set (This will find critical points), or find where • Put critical points on a number line • Plug in numbers surrounding critical points into • Plot whether it’s negative or positive. If the numbers surrounding a point go from negative to positive, it is a local minimum. If they go from positive to negative, it is a local maximum local max local min - - + +

  10. Example Find the local max/min of this function: Factor, set Find These are the critical points.

  11. Example contd. Plug in points surrounding the critical numbers into Now determine which is a local min and which is a local max. Is a local min Is a local max

  12. How to find local max/min on graph: Local max: changes from +/- Local min: changes from -/+

  13. Let’s take a quiz!

  14. Question #1 Identify the absolute extrema and relative extrema of the following function on the interval Relative max: Relative min: Relative max: Relative min: Relative max: Relative min: Relative max: Relative min: Hint

  15. Incorrect! Try again! Back to question Hint

  16. Correct! Good job! Solution Back to question Next Question

  17. Question #1: Hints • Remember that if you are given the graph of the local max is the largest y value and the local min is the smallest y value on a smaller interval of the domain. • There can also be more than one local min/max. Attempt this question till you get it right, then you can view the solution! Back to Question

  18. Question #1 Solution: • Cosine has extrema that occur at many points. It has both relative and absolute maximum of 1 at • Cosine also has both relative and absolute minimums at Move on to the next question

  19. Question #2 If on , determine the local minimum of f. A C B D Back to Question #1 Hint

  20. Incorrect! Try again! Back to question Hint

  21. Correct! Good job! Solution Back to question Next Question

  22. Question #2 Hints If you didn’t get it right, look through these hints! • Sketch a graph to find where • Utilize your knowledge of trig! Back to Question #2

  23. Question #2 Solution Move on to next question So local minimum is

  24. Question #3 If , determine the local maximum of f. A C B D Back to question #2 Hint

  25. Question #3 Hints If you didn’t get it right, look through these hints! • Make sure you find the critical point from • not Back to question #3

  26. Question #3 Solution Plug in numbers surrounding critical numbers into f’(x) and determine whether they are positive or negative Find the derivative and set it equal to 0 -2 0 2 Solve for x and plot numbers on a number line 0 is the local max. Move on to next question

  27. Correct! Good job! Solution Back to question Next Question

  28. Incorrect! Try again! Back to question Hint

  29. Question #4 Find the local min of A C B D Hint Back to Question #3

  30. Question #4 Hints: If you didn’t get it right, look at these hints and try again! • Because there are so many critical points, make sure you are testing them in on the number line Back to question #4

  31. Question #4 Solution Local min at x=4; this is where the slope (f’(x)) changes from negative to positive Find the derivative Simplify - + - + Set the numerator and denominator equal to zero, and factor if needed -2 1 4 Solve for x, Then set the points on a number line

  32. Correct! Good job! Solution Back to question END!

  33. Incorrect! Try again! Back to question Hint

  34. You’ve mastered local max/min. CONGRATULATIONS!

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