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Local Maximum and Minimum

Local Maximum and Minimum. Objectives. Students will learn how to find the local min/max through the use of a graph and algebraically (with a number line). Prerequisite Concepts. Distinguish between absolute maximum and minimum. Understand where to find absolute maximum and minimum on a graph.

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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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