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# Class Note for MATH 1431 with Professor Caboussat at UH

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This 30 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15

Definition 0 A function f is said to take on a local maximum at c if fc 2 fx7 for all X suf ciently close to c o A function f is said to take on a local minimum at c if fc fx7 for all X suf ciently close to c The local maXima and minima are called local extreme values of f f maximum Ioca l mimmum V local minimum l 1quot x2 x3 x4 1 If f takes on a local maximum or minimum at c then either fc 07 or fc does not exist Definition The numbers 6 in the domain of the function f for with either fc 07 or fc does not exist7 are called the critical numbers of f First Derivative Test Suppose that c is a critical number of f 0 If f X gt O for all X lt 6 close to c and fX lt O for all X gt 6 close to c then fc is a local maximum 0 If f X lt O for all X lt 6 close to c and fX gt O for all X gt 6 close to c then fc is a local minimum 6 If f X keeps constant sign close to c fc is not a local extreme value Second Derivative Test Suppose that f c O and f c exists 0 If f c gt 0 then fc is a local minimum 0 If f c lt 0 then fc is a local maximum 0 If f c 0 the test fails METH 1411 9 fx 7X4 6 0 METH 1411 Find and classify the critical numbers for the function The graph of f x is as follows Classify the critical numbers for f METH 1411 44 Endpoints and Absolute Extreme Values METH 1411 rmeH 1411 endpoint 7 endpoint l endpoint i i i i i i i b maximum Iminimum minimum i a b a domain a b domain 2 b endpoint I endpomt mimimum i lmaXImum i a 17 domain nm domain w b Endpoints Definition If c is an endpoint of the domain of 1 then f is said to have an endpoint maximum at c if fc 2 fx for all X in the domain of f suf ciently close to c f is said to have an endpoint minimum at c if fc fx for all X in the domain of f suf ciently close to c METH 1411 Chap 4 44 What About Absolute Extreme Values Suppose f is continuous on 37 b and differentiable on 37 b 0 Compute fa and fb 0 Compute fc for every critical number c in 37 b 0 Compare these values inliTH 1431 1 Maw 1411 1 Find the absolute extreme values for the function 1 1 fx 7 X3 X2 6X7 3 on the interval 741 Definition A function f is said to have an absolute maximum at d if fd 2 fx for all X in the domain of f f is said to have an absolute minimum at c if fd fx for all X in the domain of f METH 1411 Find the absolute extreme values for the function fXX272le2 on the interval 712 Find the absolute extreme values for the function 1 1 fx 7 X3 X2 6X7 3 on the interval 74700 Behavior when X gt ioo 7 fXgtOO aSXHOO fX becomes arbitrary large when X increases without bound fXgtOO aSXHOO fX becomes arbitrary large negative when X increases without bound Examples Behavior when X gt ioo 7 fXgtOO aSXHioo fX becomes arbitrary large when X decreases without bound fX H700 aSXH foo fX becomes arbitrary large negative when X decreases without bound Examples Polynomials 1 fx aan an71xrquot 31X 30 fxx4ix3xzxl ax f 7 2N fx X3 7 sinx rmeH 1411 Recipe extended 9 Compute the value of the functions at the endpoints if any 9 Compute the behavior of the function when x a ioo if needed 0 Compute fc for every critical number 6 in 37 b 0 Compare these values Find the absolute extreme values for the function on the interval 17 oo Find the absolute extreme values for the function fXX272X1 on the interval 7007 O MATH 1431 19328 Alexandre Caboussat Lecture MWF 11AM 12PM Office hours MWF 12PM 1PM httpwwwmathuheduNcaboussatmath1431 J HWTH 1431 1

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