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Local Extreme Points

Local Extreme Points. Objectives. Students will be able to Find relative maximum and minimum points of a function. First-Derivative Test for Local Extrema. Suppose c is a critical point for y = f ( x )

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Local Extreme Points

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  1. Local Extreme Points

  2. Objectives Students will be able to • Find relative maximum and minimum points of a function.

  3. First-Derivative Test for Local Extrema Suppose c is a critical point for y = f(x) • If f’ (x) > 0 throughout some interval (a, c) to the left of c and f’ (x) < 0 throughout some interval (c, b) to the right of c, then x = c is a local maximum point for the function f. AND

  4. First-Derivative Test for Local Extrema Suppose c is a critical point for y = f(x) • If f’ (x) < 0 throughout some interval (a, c) to the left of c and f’ (x) > 0 throughout some interval (c, b) to the right of c, then x = c is a local minimum point for the function f. AND

  5. First-Derivative Test for Local Extrema Suppose c is a critical point for y = f(x) • If f’ (x) > 0 (or f’ (x) < 0) throughout some interval (a, c) to the left of c and throughout some interval (c, b) to the right of c, then x = c is not a local minimum point for the function f.

  6. Second Derivative Test Let f be a twice differentiable function in an interval I, and let c be an interior point of I. Then • if f’ (c) = 0 and f’’ (c) < 0, then x = c is a strict local maximum point. • if f’ (c) = 0 and f’’ (c) > 0, then x = c is a strict local minimum point. • if f’ (c) = 0 and f’’ (c) = 0, then no conclusion can be drawn.

  7. Example 1 Find the locations and values of all local extrema for the function with the graph

  8. Example 2 Find the locations and values of all local extrema for the function with the graph

  9. Example 3 Suppose that the graph to the right is the graph of f’ (x) , the derivative of f(x). Find the locations of all relative extrema and tell whether each extremum is a relative maximum or minimum

  10. Example 4 Find the critical points for the function below and determine if they are relative maximum or minimum points or neither.

  11. Example 5 Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

  12. Example 6 Find the critical points for the function below and determine if they are absolute maximum or minimum points or neither.

  13. Example 7 For the cost function and the price function find • the number, q, of units that produces a maximum profit. • the price, p, per unit that produces maximum profit. • the maximum profit, P.

  14. Example 8 Suppose that the cost function for a product is given by find the production level (i.e. value of x) that will produce the minimum average cost per unit .

  15. In Summary To find local extrema, we need to look at the following types of points: • Interior point in an interval I where f’ (x) = 0 • End points of I (if included in I) • Interior points in I where f’ (x) does not exist

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