Determine Intervals оп Which а Function is Increasing or Decreasing

FIRST AND SECOND DERIVAТIVES TESTS NOТES .ctiv�s: • • • • • Determine intervals оп which а function is increasing or decreasing. Apply the First Derivative Test to find relative extrema of а function. Determine intervals оп which а function is сопсаvе upward or concave downward. Find апу points of inflection of the graph of а function. Apply the Second Derivative Test to find relative extrema of а function. 3.3 INCREASING/DECREASING FUNCТIONS AND ТНЕ FIRST DERIVATIVE ТЕSТ Definitions of lncreasing and Decreasing Functions А fu11ctio11 J is iпcreasiпg on an inteгval ,vhen. for any rwo numbers х 1 a11d х2 iп t11e inteгval, х 1 < х2 implies J(,,: i ) < J(x2). А fu11ctio11 f is dec1·easi11g 011 ап iпterval when, fOI· а11у two numbers х, a11d х2 iп tlle iпteгval, х 1 < х2 implies .f(x,) > J(x2). А function is increasing when, as х moves to the right, its graph moves up, and is decreasing when its graph moves down. For example, the function iп Figure 3.16 is decreasing оп the interval (-оо, а), is constant оп the interval (а, Ь), and is increasing оп the interval (Ь,оо ). .r•a 1 ТНЕОRЕМ 3.5 Test for lncreasing and Decreasing Functions t f Ье а fu11ction iliar is coot:inuous on the closed interval [а, Ь] and differentiaЬle the оре11 intervn.l (а, Ь). 1. If f'(x) > О for all х in (а, Ь), then f is in rea.sing on [а. Ь]. 2. If f'(x) < О fo1· all х in (а, Ь) then f is decreasing on [а, Ь]. З. If f'(x) = О for aJI х in (а. Ь), then f is con tant on [а, Ь]. 1 1 : Coostttnt : --1--------- z • 1 Гtxl < О : Г!ХI • О : Г{..r) > о The derlvative is related to the slope or а function. Figure 3.16 Example 1: Find the open intervals оп which f (х) answer. • :\. _ � l"'l,,1::: 3)?- '3 х => 1 = х 3 - �2 х 2 is increasing or decreasing. Use а graph to support your З);� -w� -:::t.. с, � 0 - 3х()(-\) с. с:> )(.:::о ��J lnterval Test Value Sign of f' (х) Conclusion (,- Ро I -, f' {-, (о, 1) - ':> О) ::: fo ( 1( �G0ьl\SI� = -..-=J,'::) - Dб�JJ(, 1 J-> G�.ДS 1 t., {, о�,лtс, .s f\11)�� )( :: ' :::ir} � uн� L .,U(Z. ).J t "'" Рт� • CONCAVITY AND ТНЕ SECOND DERIVATIVE ТЕSТ Definition of Concavity Let / Ье differentiaЫe on an open interval 1. The grapl1 of / is concave щнvard on 1 wben f' i increasing on the interval алd concave do, nward ор / wben /' i decrea ing on the intervaJ. The following graphical interpretation of concavity is useful. 1. LetfЬе differentiaЫe on an open interval /. lf the graph off is concave upward on /, then the graph off lies above all of its tangent lines on /. [See Figure 3.24(а).] 2. LetfЬе differentiaЫe on an open interval /. lf the graph of f is concave downward on /, then the graph off lies be/ow all of its tangent lines on /. [See Figure 3.24(Ь).] ., у r 11. (kcrea51111- Соос , oownwan1 _..,____________ Jt The graph of f lies above its tangent lines. Figure 3.24(а) The graph of t lles Ьelow its tangent lines. Figure З.24(Ь) То find the open intervals on which the graph of а functionf is concave upward or concave downward, you need to fin. the intervals on which/' is increasing or decreasing. ,,,,.,,� For instance, the graph of f(x) = ½х 3 - х is concave downward on the open interval ( -оо, О) because/'(х) = х2 -1 is decreasing there. (See Figure 3.25.) Similarly, the graph of f is concave upward on the interval (О, оо) becausef' is increasing on (О, оо). f'i dt.,c....;,.. fi,iacrt,1$ЩJ.. n1e concavity of f is related to t slope of the derivative. Figure З.25 ,... THEOREM 3.7 Test for Concavity Let f Ье а fL111ction wlюse secoпd deгivative exists on ап ореп i_nterval I . . Н f"(x) > О_ fог all х in J, then the g1·aph of f is concave upwaгd on 1. . If f"(x) < О for a\l х iл !, then the graph of f is concave downwaгd оп /. А proof of this theшem is given in Appendix А. ?" ..._____________________ (,,,,о ____, Example 3: Determine the open intervals on which the graph of f(x) downward. Use а graph to support your answer. �('{.. ::= , � (i. е -J..i­ �s ( � 6'D 2 = е-х 12 is concave upward or concave '2- := е,, � с)( J � ['1-- � ( ��( _, "t.'1-/ \ е� (-1 -t 1nterval о Test Value Sign of f"(x) Conclusion • LC>'-' c.,.�:J{;.. L' � •• Gе>юС-АNб �t..s ( l, ,,о 2-