![Definition: Local Maximum &
Minimum
Local Maximum (relative maximum)
A function f has a local maximum at c if f(c) ≥ f(
x) when x is near c. [This means that f(c) ≥ f
(x) for all x in some open interval containing c.]
Local Minimum (relative minimum)
Similarly, f has a local minimum at c if f(c) ≤ f(
x) when x is near c.](https://image.slidesharecdn.com/maximumsandminimum-150322233343-conversion-gate01/85/Maximums-and-minimum-5-320.jpg)
Maximums and minimum
- 1. Maximum & Minimum Values Calculus 1 Chapter 4.1 and 4.3 Ms. Medina
- 2. Why is this important? • An important application of differential calculus are problems dealing with optimization. Examples: • What is the shape of a can that minimizes manufacturing cost? • What is the maximum acceleration of a space shuttle? • What is the radius of a contracted windpipe that expels air most rapidly during a cough?
- 3. Definition: Absolute Maximum & Minimum Absolute Maximum (global maximum) A function f has an absolute maximum at c if f(c) ≥ f(x) for all x in D, where D is the domain of f. The number f(c) is called the maximum value of f on D. Absolute Minimum (global minimum) A function f has an absolute minimum at c if f(c) ≤ f(x) for all x in D and the number f(c) is called the minimum value of f on D. The maximum and minimum values of f are called the extreme values of f.
- 4. Find the Maximum and Minimum ? Minimum value f(a) Maximum value f(d) Absolute Minimum value at f (x) = 0 No Maximum value No Minimum value No Maximum value
- 5. Definition: Local Maximum & Minimum Local Maximum (relative maximum) A function f has a local maximum at c if f(c) ≥ f( x) when x is near c. [This means that f(c) ≥ f (x) for all x in some open interval containing c.] Local Minimum (relative minimum) Similarly, f has a local minimum at c if f(c) ≤ f( x) when x is near c.
- 6. Definition: Critical numbers Critical Numbers • A critical number of a function f is a number c in the domain of f such that either f`’(c) = 0 or f’(c) does not exist. • If f has a local maximum or minimum at c, then c is a critical number of f.
- 7. Closed Interval Method Finding the absolute maximum and minimum values of a continuous function f on a closed interval a, b: 1. Find the values of f at the critical numbers of f in a, b . 2. Find the values of f at the endpoints of the interval. 3. The largest of the values from Steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
- 8. Increasing & Decreasing Test Increasing If f’(x) > 0 on an interval, then f is increasing on that interval. Decreasing If f’(x) < 0 on an interval, then f is decreasing on that interval.
- 9. The First Derivative Test • Suppose that c is a critical number of a continuous function f. • If f’ changes from positive to negative at c, then f has a local maximum at c. • If f’ changes from negative to positive at c, then f has a local minimum at c. • If f’ does not change sign at c, then f has no local maximum or minimum at c. • Example: If f’ is positive on both sides of c or negative on both sides Direction Critical Point Direction + c - - c + - c - + c +
- 10. Definition: Concave up & Down Concave Up • If the graph of f lies above all of its tangents on an interval I, then it is called concave upward on I. Concave Down • If the graph of f lies below all of its tangents on I, it is called concave downward on I.
- 11. Concavity Test • If f”(x) > 0 for all x in I, then the graph of f is concave upward on I. • If f”(x) < 0 for all x in I, then the graph of f is concave downward on I.
- 12. Definition: Inflection Points • A point P on a curve y = f( x) is called an inflection point if f is continuous there and the curve changes from • concave upward to concave down- ward • concave downward to concave upward at P.
- 13. The Second Derivative Test • Suppose f’’ is continuous near c. • If f’(c) = 0 and f’’(c) > 0,then f has a local minimum at c. • If f’(c) = 0and f”(c) < 0,then f has a local maximum at c. Example of Second Derivative Test
- 14. Reference • Stewart, J., Calculus Early Transcendental, sixth edition. 2008.