Critical Points Of Mins And Maximum

Maxima Minim Finding Critical Points 7I A Fox X X B 3 go D hex Ecosx TzLXLTz A Ex x2 x 3 F x 3 Rewritefunction o Factor I Simplify B 963 3 4 4 3 g 12 3 12 2 12 2 x 0 12 2 2 3 3 5 5 3 0 X 0 35 0 Ei o E PIX D hex Ecosx has extsinx CEO X x 0 1 x n 0 0 1 x D Differentiate X 2 0 x TE C h x E PG 3 11 25 0 2 0 43 4 3 3 2 3 2 6 3 1 3 x2 cosxle e'Esinxt Cos 1 0 Six Cos D SEE X x 3 3 16 232 3 31 26 31 6 6 2 5 7 2 6 254 Max b Min Absolute 1 Find Criticalpoint of f 2 Evaluate f on interval ab at each Critical point in aib 3 Evaluate f at each endpoint in 4 Small valve Min A 1 2 3 x 2 3 3x X x 3 0 I I 3 2 2 X 2 033 31032 2 3 31232 3 Ft't 4213 3142 f 2 413 3145 4 6x 4 13 12 6 2 on interval 4 6 2 12 1 2 0 0 0 0 Min 1 Yg L Max 4 Max is point 4,16 B 3 2,4 IET 2 Flo f 1 b Max Large on interval a P function 1411h Is 1 2,2 12 3 12 20 11 0 X2 1 0 X 2 4 2.193 31 1885 3 4 Absolute C 2 43 Min x 3 Min E Max Max 1 3 at interval 2,24 F1 3 1 43 43 43 13 43 X 43 2 437 0 IE 2 FLO 210143 053 F Ya 2 2 3 IB EYE 4 Max D 2 3 33 3 2 2 3 o 0 1.11 SEE 42 1.19 Min 1684 23,8 1 6 2 66 168 6 x 11 33 a 1 3 7 2 o Dec Inc functions b a function f is increasing on an interval I in I andX Lx a function f is decreasing on aninterval I First Derivative Test if fix for all a fax x if Fox fax for an x ex tx in I a x L xz Suppose cfs is on an continuous on a closedintervalcarb r differtiable openinterval Lab 1 if F x O for all x in Ca b then f is increasing on a b 2 if f x 20 for all x in Ca b then f is decreasing on Ea b 3 if f x 0 For all x in Ca b the Cfs is Constant on the interval y Y 4 Fox Ej whenever x2 X Whenever gg.am Xz x I I I To Find Interais of Inc at one test valve in each interval Determine sign of f 1 f 5 x 3 x2 0 2 B F XC x 0 2 XI 10.211401 1 I 3 in c del the x2 x 1 x2 3 3 e E x 1 0 e I X 3 g p p dec in 3 x 2x E F µ 2x X iii Test b Del dec x2 X 3_ 2 33 0 Applying first derivative test to find local extrema Find the local 1 FIX 4 Sign 2 2 x 4 3 0 inc 3 dee 4 2x2 4 4 X 244 312 dec x4 extrema or 1 24 Inc 0 2 X 1 0 94 1 g 3y x x 1 3 31.6231241 3121 35124 3 3121 351 3,12 six 2 x2 723 31 14 III Test_ 27 2523 dee 43 2543 inc in Ethan 3 local min If 2 19 3 3 0,0 73 Concavity of a function and the Second derivative a function f afunction f is Concavedownward on an interval I 1 if P SO then f 2 if f Lo f is Concave upward on an interval I if f then is concave f o if f is dec f co upward is concave is inc te on downward I on I t 1 find X valves at which f determine test interval of 2 Determine the 3 use test to sign determine x o or of f if f x due use points to atone testvalve in each interva whether fi s concave up or down D Changes Concavity at C x f Points of Inflection A point e f les is called graph f or point of inflection of the F CX DNE at X c and f a Find theintervals of concavity and thepoints f X 2 63 x graph 1 Determing x valves at which F Cx o FX 2 67 X P x 82 4 3 0 36 12 2 f x 12 13 124 X of inflection of the of 0 EET o 3 x F Cx DNE Concave up i X 013 ye 0,3 t.t.si me at f 3 83 Points Of inflection 10,0 P 3,83 I 1 f 2 5 x 4 10 3 5 F x 20 3 305 f x 60 x t 60 60 1 1 It 3E 20 E g F G IS t f lx 60 t I E 30 E 70 SOE t 90 E 60 E t 4E 240 t 3 t 180 t Got O EYE THE Etten IEIE.IEE Apply Second derivative test to find local extreme P 1 If c 2 If f SO then Itamin flicks at each critical point minimax Usesecond derivate to find extremas of f 4 1 2 P X f 3 f 2 3 4 3 o 4 PEE B FIX 1 f x F 2 4 4 3 12 24 4 2 x x P O f C2 43 8 16 4,1 16 y Min 0 Max CO 4 128 c 3 1 x'ex f X X ex ex 2x O x2 2x ex O x 2 ex O x2 2x ex F x 164 Min 80 16 X 4 0 2 12 O X e 2 F x 8 2 12 X P 3 x4 4 x fees then test fails Evaluate 1 F P min then f has local maximum b Max FCC c 20 0 c 3 If F f has local minimum X O Ze 2x ex ex 2 E 2 2 Max x 4 2 e O O min 2 C 2 15 X X't 15 1 F X 3 F 2 f 16 1 2 3 min 840 1 813 fix 7 P x 32 f 2 2 8 X X I Max 2 6 51 2x 64 512 851 X 2 30 x 10,516 8 X 2 30 x O 16 x 3 x 30 x 6 X I D Ya F 16 10 x X 2 3 Ey 8 64 512 64 16 I 8,888 minx E 2 A 1 fCxJ x3 X 13 3 2 x 13 11 X f x III I 6 2 F ax 9 26 six K x 3 11 2 26 9 8 1 74 Graphing Functions Info needed to analyze and sketch a graph 1 Domain 2 X b y intercepts 3 Asymptotes 4 Critical points p local 5 of Points Analyze A f 3 1 Domain 2 X inflection and sketch x 3 s All Reall yjfqg by.int 3 In of polynomial 3 x2 014h0m1al Always 9 y O 0 no x 3 Polynomials dont have them Lone m graph x2 X Set11 0 Solve Ya SetX O Solve 3 extrema x 0 XI to 5 Points of inflection 213 Ii I offing sign Mom F f 2 x 1 43 i 0 2 3 1 2 X 4 I 2 3 Analyzing A and Sketching fix 1 Domain 2 xp y int graph of a a rational 31E X 4 Y 3 Asymptotes I 2 X 0 Allreaisexceptt 3841,1 34481 x m 362 0 81 IF I 34 252 Fox X 341,1 x 41161,531 2 0 EYE 2 01 641 345 3 3 8112 2 0 Aegis 12.82 2,1 Horizontal Asymptotes F r 9g vertical Asymptotes x x2 0 the Tmz 3 4 function o 0 3 F'CX 2 F'Cx x2 4446434 5 o 3 2x 270 Point of inflection f 4 I i Sign F 7212 f 4 24 144 f x o X is undefined at X 12 but X 12 are not in the domain of the function fax therefore f has nopointof inflection