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Understanding Extrema of Functions on Intervals

Learn the definitions of extrema on intervals, finding relative extrema, critical numbers, and guidelines for identifying extrema on closed intervals in calculus.

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Understanding Extrema of Functions on Intervals

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  1. EXTREMA ON AN INTERVAL Section 3.1

  2. When you are done with your homework, you should be able to… • Understand the definition of extrema of a function on an interval • Understand the definition of relative extrema of a function on an open interval • Find extrema on a closed interval

  3. EXTREMA OF A FUNCTION

  4. DEFINITION OF EXTREMA Let f be defined on an open interval I containing c. • is the minimum of f on Iif for all xin I. 2. is the maximum of f on Iif for all xin I.

  5. EXTREMA CONTINUED… • The minimum and maximum of a function on an interval are the extreme values, or extrema of the function on the interval • The singular form of extrema is extremum • The minimum and maximum of a function on an interval are also called the absolute minimum and absolute maximum on the interval

  6. EXTREMA CONTINUED… • A function does not need to have a maximum or minimum (see graph) • Extrema that occur at endpoints of an interval are called endpoint extrema

  7. THE EXTREME VALUE THEOREM If f is continuous on a closed interval then fhas both a minimum and a maximum on the interval.

  8. DEFINITION OF RELATIVE EXTREMA • If there is an open interval containing c on which is a maximum, then is called a relative maximum of f, or you can say that f has a relative maximum at • If there is an open interval containing c on which f is a minimum, then is called a relative minimum of f, or you can say that f has a relative minimum at

  9. Find the value of the derivative (if it exists) at the indicated extremum. • 0.0 • 0.0

  10. Find the value of the derivative (if it exists) at the indicated extremum. • 0.0 • 0.0

  11. Find the value of the derivative (if it exists) at the indicated extremum.

  12. DEFINITION OF A CRITICAL NUMBER Let f be defined at c. • If , then c is a critical number of f. • If f is not differentiable at c, then c is a critical number of f.

  13. Locate the critical numbers of the function. • None of these

  14. Locate the critical numbers of the function. • None of these

  15. THEOREM: RELATIVE EXTREMA OCCUR ONLY AT CRITICAL NUMBERS If f has a relative maximum or minimum at , then c is a critical number of f.

  16. GUIDELINES FOR FINDING EXTREMA ON A CLOSED INTERVAL To find the extrema of a continuous function f on a closed interval , use the following steps. • Find the critical numbers of in . • Evaluate f at each critical number in . • Evaluate f at each endpoint of . • The least of these outputs is the minimum. The greatest is the maximum.

  17. The maximum of a function that is continuous on a closed interval at two different values in the interval. • True • False

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