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This 5 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH115 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/215398/math115-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.
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Date Created: 09/28/15
A function of two variables has a local maximum at abif fxyS fabfor all points xyin some region around ab n 39 mlwvwo axx llltf39 o o Q o o 2 lum39owwv 10 c u 0 outside the region It Is 7 II W I 3 possible that the function could 2 030 7 392 be larger A function of two variables has a local minimum at abif fxy2 fabfor all points xyin some region around ab 0 outside the region it is possible that the function could be smaller A point 5119 is called a critical point of f is one of the following is true i Vfab 0 that is BOTH fxab 0 and fyab 0 ii fx ab and or fy abdoesn39t exist f has a local maximum or fxab 0 and fyab 0 local minimum at 5119 and the rst partial j derivatives of f exist there abis a critical pomt f has a local maximum or localminjmum at a b a b1s a critical pomt not all critical points lead to a local maximum or local minimum f00 0 maximum in the the graph is in the direction of the xaxis Shape Of a saddle minimum in the the point 000 is direction of the yaxis called a saddlepoint Find all critical points absuch that Mm 0 and fyab 0 Let f m f and D quoty fmfy fw2 the second partial derivatives are fxy fw contiuous in some region around ab Evaluate D at thesecritical points fxx a 7 gt Ow mb s alocal miniman Dab gt 0 f a b lt OI fabis a local maximum Da lt 0 W mb s asaddle point Da OWthe test gives no information fxy3x2y313 323 3312 2 fx 6xy6x f 3x23y2 6y fx6xy1 fl 0either6x001y l0 ax0fy 3y2 6y 3yy 2 3 y00ry2 00and 02 by1fy 3x23 6 3x2 1 3xlorx l 11 and 11 fxy3x2y313 323 3312 2 f 6xy6x f 3x2 3y2 6y fn6y6 fw6y6 D0036 m fly 6x D6y626x2 Dll 36 2 2 doesn39t D 36y 1 x fm l matter D f 002 gt 0 lt 0 02 2 gt 0 gt 0 110 lt 0 110 lt 0 D02 36 fu02 12 D 11 36 doesn39t aemk matter Classification local max local min saddle pt saddle pt A function of two variables has an absolute maximum at abif fx y S fabfor all points in the domain of f A function of two variables has an absolute minimum at abif fx y 2 fabfor all points in the domain of f absolute maximum Usually the domain is restricted to some region fxyx2y2x2y4 Restricted Domain leSland lSySl absolute minimum A region in R2 for us this will be the xy plane is called closed if it includes its boundary A region in R2 for us this will be the xy plane is calledbounded if it is contained within some disk in other words a region is bounded if it is finite Extreme Value Theorem there are polnts a band c d1n the reglon S fx y ls commuous In some 2 so that fabis an absolute maximum and closed bounded region S in R2 fcdis an absolute minimum This tells us that the points exist but it doesn t tell us how to find them To find the absolute maximum and absolute minimum values of a continuous function f on a closed region S 1 Find all the critical points of f that lie in the region S Evaluate the function at each of these points 2 Find all extreme values of f that lie on the boundary This turns into a Calc I problem 3 The largest and smallest of the values found in steps 1 and 2 are the absolute maximum value and absolute minimum value of the function f Find the absolute maximum and absolute minimum values of f x y x2 xy on the region S x yx S 2 yl S 1 L 2 1 Find possible critical pts inside the region Q fxyx2xy 11 D5 la set S Critical t 39 quot fx2xy fyx0 p Both have to be true at the same time 0 0 L 3 4 plugging in x 0 intofx gt y 0 2 Find all extreme values off that lie on the boundary 2 0 INF 2 f 2 y 2 2 ef ng 2 on 7393 b iny1 fxa1x2x gtfonL2x2xt 71 1 f onL22cl0gtc 71 2 W2 Ml 2y 222323 61 L41y 1 fx 1x2 x gtf onL4x2 x 1 f onL42x l gtx f000 f 711 71 f 711 71 absolute maximum absolute minimum absolute minimum