Notes: Critical Numbers, Intervals of Increasing/Decreasing, Curve Sketching Day 2 Intervals of Concavity, & POI Example One: Use the graph of f to find the following: Intervals of Increasing: Decreasing: Maximum: Minimum: Intervals of Concave Upward: Concave Downward: Critical points: A number 𝑐 in the domain of 𝑓 is called a critical point If: 𝑓′(𝑐) = 0 𝑜𝑟 𝑓′(𝑐) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 Then: 𝑥 = 𝑐 is a critical point. Increasing/Decreasing Behavior of Function If 𝑓′(𝑥) > 0, then 𝑓 is ____________ If 𝑓′(𝑥) < 0, then 𝑓 is ____________ Concave Up/Concave Down Behavior of Function If 𝑓′′(𝑥) > 0, then 𝑓 is ____________ If 𝑓′′(𝑥) < 0, then 𝑓 is ____________ Points of Inflection: A number 𝑐 in the domain of 𝑓 is called a point of inflection If: 𝑓′′(𝑐) = 0 𝑜𝑟 𝑓′′(𝑐) = 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 And: If sign change at c Then: 𝑥 = 𝑐 is a point of inflecton

Critical

Numbers

AD

How does the sign of the first derivative

relate to the original function?

AD

How does the sign of the second derivative relate to the original

function?

AD

AD

Points of Inflection

Graph of 𝑓(𝑥)

÷i÷÷÷÷÷÷÷÷÷÷¥eg* .'

thereissigncnangeatx.cl#x--cisqxmtraqqnmain,increasingdecreasing

.¥ Ton:c:÷..

Notes: Critical Numbers, Intervals of Increasing/Decreasing, Curve Sketching Day 2 Intervals of Concavity, & POI Example Two: 𝑓(𝑥) = 𝑥3 − 27𝑥 − 20 Critical Numbers: Intervals of Increasing: Decreasing: Maximum Value: Minimum Value: Possible Points of Inflection: Intervals of Concave Upward: Concave Downward: Point(s) of Inflection: Example Three: 𝑓(𝑥) = 1

3𝑥3 − 𝑥2 + 𝑥

Critical Numbers: Intervals of Increasing: Decreasing: Maximum Value: Minimum Value: Possible Points of Inflection: Intervals of Concave Upward: Concave Downward: Point(s) of Inflection:

1-75×9,39 f' Cx)-÷#PHX) ✓A @ V flex) V

X=I3flax

's-27×-20

to , -37013,0 )f' Cx) --3×727 f'CX)-3×2-27 FIX) -- X

'-27*-20

0=3×2-27 f- "CX)=6X f(37=(373-2713) -20( - 313734 3×2=27 0=64 = 27-81-20=-74

{-33,1-74} F- 9 KO ff-37=1-313-271-3) -20X-- 13 f'' (x)=6X = -271-81-20=34

X=O f' CX) -- 3×2-27 f'' fl) - mg( O , ) f 't-4) -- Pos fucpjpos flo)= - 201- 0,0 ) f' co)

-- neg

( O,-20)

f' (4) -- pos

^

Fit fix) )-Ign FIX)(

* IV Va -

l -Dib) text. }x3×2+× fYX)=xZ2xHnone

}none f-'CX) -1341×4-2×+1 f'4×7=2×-2

74×7=112-2×+10=2×-2

XII XII( 1,07 0=1×-174-1)C- as,l ) x -- I f- "CX) -- 2×-2

( 1,43 ) f' (x) - ( X- 1) (X - l) f'407 -_ neg

fatten.ci/txfifzf--=hpe9fYp9s3--P0Sf'Y23--pos--pos

Notes: Critical Numbers, Intervals of Increasing/Decreasing, Curve Sketching Day 2 Intervals of Concavity, & POI Example Four: 𝑓(𝑥) = 3𝑥5 − 5𝑥 + 1 Critical Numbers: Intervals of Increasing: Decreasing: Maximum Value: Minimum Value: Possible Points of Inflection: Intervals of Concave Upward: Concave Downward: Point(s) of Inflection: Example Five: Where does 𝑓(𝑥) have critical numbers? Where is 𝑓(𝑥) increasing? Where is 𝑓(𝑥) decreasing? Are the critical values local minimums, maximums, or neither? Where is 𝑓(𝑥) concave upward? Where is 𝑓(𝑥) concave downward? Where does 𝑓(𝑥) have points of inflections? What is the difference between critical numbers and points of inflection?

Graph of 𝑓′(𝑥)

MmaPI41z9f4X7_pEi@pfCx7Aqq-Ahrq9fYx7toyoy4z.o) HH

-

- 3*5×4+1 fix --I5x4. 20×3(0,413) thx) -- 15×4-20×3 f " 60×3-60×210,17 0=5×313×-4)

0=60×4×-1)( 413, ) 5×3=0 3×-4=0

60×2=0 X- 1=0X=O&X=l F- O E- 413 KO XII11,0 ) f ' = 5×313×-4) f " 60×4×-1)to,Dflfp=negCneg7=t

nxg¥ Cl , -17 f' Citroen = - tf't:{'¥÷mg§cfg!§÷IAOKI f' (2)

'

- poscpos)=tf- (Iz) -- some#

f- (1) =3 -5H= LMAIWAYSf'Cx) (x- intercepts) make af- 'CX) > 0 (above x-axis) sign linef- ' (x) CO (below X- axis) if giveng- a graphof

11=0,8 , & 13 &derivative

• • •

( O , 8) U ( 13,0 ) flick) -- Slopes*

of flex)8-

C-0,0 ) U ( 8,13 )fiYX7-oatfYX1max1minIomiFm@I131f4x7mmairf.o: # 08*-13, • Hx)

x--8 FLATFEET'.FI#9f' 'Cx)C- P, 2714,6741 , )

④ ④nothing

(214716,11 )

11=2,4 , 6,811-

fi nodfefinedtttiundefindAndy must have sign change