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# DIFFERENTIAL CALCULUS MTH 251

OSU

GPA 3.79

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This 2 page Class Notes was uploaded by Mrs. Dedric Little on Monday October 19, 2015. The Class Notes belongs to MTH 251 at Oregon State University taught by Staff in Fall. Since its upload, it has received 9 views. For similar materials see /class/224444/mth-251-oregon-state-university in Mathematics (M) at Oregon State University.

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Date Created: 10/19/15

Sec 41 Maximum and Minimum Values Calculus provides effective power means for solving a host of optimization problems such as maximizing the pro t of a company or minimizing the pollution of an industrial process Maximum and Minimum Values M is the global maximum of a function f on a set D if fc M for some 0 in D and S M for all x in D m is the global minimum of a function f on a set D if fd m for somedin D andfx 2 m for allx in D Closely related but different concepts follow M is a local maximum of a function f if f c M for some 0 and f S M for all so near 0 m is a local minimum of a function f on a set D if f d m for some d and f 2 m for all so near d The statement for all so near 0 means there is an open interval I that contains 0 such that f S M for all x in I Consequently if the domain of f is the closed interval 11 then neither f 1 nor f b is a local max or min however either may turn out to be the global max or min This subtle distinction is important for what follows The following observation also is important If the domain of f is the closed interval 1 b and if either the global max or min occurs at a point inside 11 the the global max or min is also a local max or min MaxMin Theorem If fx is continuous on a closed interval ab then f takes on its maximum value M and its minimum value in at certain points in 1 b that is there are points 0 and d in the interval 1 b such that mfd f f0M for all so in 11 Note Well If either of the words continuous or closed is omitted from the statement of the Max Min Theorem then the statement is false The max min theorem tell us that maximum and minimum values are waiting to be found at least for a function f continuous on a closed interval 11 but the theorem does not tell you how to nd the maximum and minimum values That is where calculus come into play On Locating Extreme Values AKA local or global maximum or mini mum values Theorem 1 If f has a local max or min at c then either f 0 0 or f c does not exist De nition Let f be de ned for all so near 0 The c is called a critical point number of f if either f c 0 or f 0 does not exist With this language Theorem 1 can be expressed as Theorem 1 If f has a local max or min at c then 0 is a critical point of The importance of this result for us is that we can limit the search for points where f may have a local max or min to the critical points of Because of an earlier remark which one nding critical points also helps us nd global extreme values Indeed the foregoing leads to the following procedure for nding global extreme values The Closed Interval MaxMin Method Let f be continuous on a closed interval 11 Step 1 Find f a and f Step 2 Find f c for all critical points 0 of f in 11 Then the global max respectively min of f is the largest respectively small est of the numbers found in Steps 1 and 2 Why or when is this method likely to be helpful

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