8/4/2019 wikimama Class 12 Ch 06 Application of Derivatives

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FSSINMF\NHK HA LOZNQF\NQO

Loznqf|nqo n} szndfzni~ u}ol |h lo|ozdnko zf|o} ha m`fkbo} ha vufk|n|no}+ Loznqf|nqo iofl} |h |`o

lo|ozdnkf|nhk ha }ihso ha |`o |fkbok| |h f muzqo f| f bnqok shnk|+ Loznqf|nqo fi}h `ois} nk ihmf|nkb

|`o |uzknkb shnk|} ha |`o bzfs` ha f aukm|nhk+

Bohdo|znmfi Dofknkb ha |`o Loznqf|nqo5

Io| ~ 4 a (p. zoszo}ok| f bnqok muzqo) fkl S (p) ~. fkl

V(p ' p) ~ ' ~. jo |rh ln}|nkm| fkl konb`jhuznkb

shnk|} hk |`o muzqo+ Io| |`o m`hzl SV) ehnknkb S fkl

V) dfgo fk fkbio rn|` |`o p$fpn} fkl |`o |fkbok|

|h |`o muzqo ~ 4 a(p. f| |`o shnk| (p) ~. dfgo fk fkbio

rn|` |`o sh}n|nqo p $fpn}+ \`o }ihso ha |`o m`hzl SV

n} bnqok j~

|fk 4VZ ~ a(p p. a(p.

SZ p p+

H

~ 4 a(p.

S (p) ~.

~

p

Z

V(p' p)~' ~.

~

p

\fkbok| fkl Khzdfi |h f Muzqo5

S\ n} |`o |fkbok| |h |`o muzqo ~ 4 a(p. f| |`o shnk|

S(p6) ~6.+ SK n} |`o khzdfi |h | o muzqo f| S+ \`o

khzdfi |h |`o muzqo f| shnk| S n} sozsoklnmuifz |h

|`o |fkbok| |h |`o muzqo f| S+

\`o }ihso ha |`o |fkbok| f| S(p6) ~6. n})

6 6(p )~ .

l~

lp+

\`o }ihso ha |`o khzdfi f| S(p6) ~6. n})

6 6(p )~ .

6

l~lp

+

~ 4 a(p.

S (p6 ) ~6 .

>33

\ K

>3 3 '

@okmo |`o ovuf|nhk ha |`o |fkbok| S\ n})

6 6

6 6(p )~ .

l~~ ~ (p p .

lp) fkl |`o ovuf|nhk ha |`o

khzdfi SK n})

6 6

6 6

(p )~ .

6~ ~ (p p .

l~

lp

4 (p p6. ' (~ ~6.l~

lp 6 6p )~4 3+

\`o ovuf|nhk} ha |`o |fkbok| fkl |`o khzdfi fzo

b"(|.

b(|. (p a(|.a"(|.~ . fkl a "(|. p a(|. b"(|. ~ b(|. 3 +

Fkbio Jo|rook |rh Muzqo}

8/4/2019 wikimama Class 12 Ch 06 Application of Derivatives

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:

\`o fkbio jo|rook |rh muzqo} (hz |`o fkbio ha

nk|oz}om|nhk ha |rh muzqo}. n} loankol f} |`o fkbio

jo|rook |`o |rh |fkbok|} f| |`onz shnk| ha

nk|oz}om|nhk+ F} |`o anbuzo }`hr}) ) |`o fkbio

jo|rook |`o |rh muzqo}) n} bnqok j~ 4 6 $ :

|fk 4 |fk ( 6 $ :.

4 6 :

6 :

|fk |fk)

6 |fk |fk

r`ozo |fk 6 4 a"(p6. fkl |fk : 4 b"(p6.+

~ 4 b (p.

S( p6) ~6.

~ 4 a(p.

H \:( .p p6

\6

~

p

\rh muzqo} fzo }fnl |h mu| ofm` h|`oz hz|`hbhkfii~ na |`o fkbio jo|rook |`od n} f znb`| fkbio)

|`f| n}) na 4 >33 ) nk r`nm` mf}o ro rnii `fqo) |fk 6 |fk : 4 $6+

\rh muzqo} |hum` ofm` h|`oz na |`o fkbio jo|rook |`o |fkbok|} |h |`o muzqo} f| |`o shnk| ha

nk|oz}om|nhk n} 3h ) nk r`nm` mf}o ro rnii `fqo) |fk 6 4 |fk : +

Dhkh|hkhmn|~

Io| ~ 4 a (p. jo f bnqok aukm|nhk rn|` L f} n|} lhdfnk+ Io| L6L +

Nkmzof}nkb Aukm|nhk5

a (p. n} }fnl |h jo nkmzof}nkb nk L6 na ahz oqoz~

p6) p: L6) p6 ; p: a(p6. ; a (p:.+ N| dofk} |`f| |`o

qfiuo ha a (p. rnii goos hk nkmzof}nkb rn|` fk

nkmzof}o nk |`o qfiuo ha p+ Zoaoz |h anb+:+

~

p

Anb+ :Nkmzof}nkb Aukm|nhk}

H

Khk$Lomzof}nkb Aukm|nhk5

a (p. n} }fnl |h jo khk$lomzof}nkb nk L6 na ahz oqoz~

p6 ) p: L6 ) p6 ; p:a (p6. a (p:.+ N| dofk} |`f| |`o qfiuo ha

a (p. rhuil koqoz lomzof}o rn|` fk nkmzof}o

nk |`o qfiuo ha p+ Zoaoz |h anb+=+ pAnb+ =Khk Lomzof}nkb Aukm|nhk}

H

Lomzof}nkb Aukm|nhk5

a (p. n} }fnl |h jo lomzof}nkb nk L6 na ahz oqoz~

p6) p: L6) p6 ; p:

a (p6. 9 a (p:.+ N| dofk} |`f| |`o

qfiuo ha a (p. rhuil lomzof}o rn|`

fk nkmzof}o nk |`o qfiuo ha p+ Zoaoz |h anb+

8/4/2019 wikimama Class 12 Ch 06 Application of Derivatives

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a (p. n} }fnl |h jo khk$nkmzof}nkb nk L6 na ahz oqoz~

p6) p: L6) p6 ; p:

a (p6. a (p:.+ N| dofk} |`f| |`o qfiuo ha a (p.

rhuil koqoz nkmzof}o rn|` fk nkmzof}o nk |`o

qfiuo ha p+ Zoaoz |h anb+ ?+

~

pH

Anb+ ?Khk Nkmzof}nkb Aukm|nhk}

Kh|o5 F aukm|nhk r`nm` n} on|`oz nkmzof}nkb hz lomzof}nkb nk |`o ok|nzo nk|ozqfi L6 n} mfiiol f

dhkh|hknm aukm|nhk+

Jf}nm \`ohzod}5

Io|) a jo f aukm|nhk |`f| n} mhk|nkuhu} nk Wf) jT fkl lnaaozok|nfjio nk (f) j.+ \`ok

(n. a(p. n} f khk$lomzof}nkb aukm|nhk nk Wf) jT na a "(p. 3 nk (f) j. 7

(nn. a(p. n} fk nkmzof}nkb aukm|nhk nk Wf) jT na a "(p. ; 3 nk (f) j. 7

(nnn. a(p. n} f khk$nkmzof}nkb aukm|nhk nk Wf) jT na a "(p. 3 nk (f) j. 7

(nq. a(p. n} f lomzof}nkb aukm|nhk nk Wf) jT na a "(p. 9 3 nk (f) j.+

Zodfzg}5(n. Io| p6) p:) (p6 9 p:. jo fk~ |rh shnk|} nk Wf) jT+ ]nkmo |`o aukm|nhk n} lnaaozok|nfjio fkl

mhk|nkuhu} nk Wf) jT) a(p. }f|n}ano} |`o mhkln|nhk} ha Ifbzfkbo} Dofk qfiuo |`ohzod hk Wf)

jT+ @okmo hk Wp6) p:T)a(p:. a(p6. 4 (p: p6. a (m.) p6 9 m 9 p:+ ]nkmo) p: p6 ; 3) a(p:.

; a(p6. na a (m. ; 3 fkl a(p:. 9 a (p6. na a (m. 9 3+

\`n} zo}ui| n} |zuo ahz fk~ |rh shnk|} p 6 fkl p: nk Wf) jT fkl m (p6) p:.+ ]nkmo m n}

fzjn|zfz~) ro mhkmiulo |`f| na a (p. ; 3 nk Wf) jT) |`ok a (p. n} fk nkmzof}nkb aukm|nhk fkl na

a (p. 9 3 nk Wf) jT) |`ok a(p. n} f lomzof}nkb aukm|nhk+

(nn. Na a "(p. 3 p (f) j. fkl shnk|} r`nm` dfgo a " (p. ovufi |h ozh (nk jo|rook (f)

j. lhk| ahzd fk nk|ozqfi) |`ok a (p. rhuil jo nkmzof}nkb nk Wf) jT+

(nnn. Na a "(p. 3 p (f )j. fkl shnk|} r`nm` dfgo a " (p. ovufi |h ozh (nk jo|rook (f)j. lhk| ahzd fk nk|ozqfi) a (p. rhuil jo lomzof}nkb nk Wf) jT+

(nq. Na a (3. 4 3 fkl a "(p. 3 p Z) |`ok a (p. 3 p ($ ) 3. fkl a (p. 3

p (3) .+

(q. Na a (3. 4 3 fkl a "(p. 3 p Z) |`ok a(p. 3 p ($ ) 3. fkl a (p. 3

p (3) .+

(qn. \`o shnk|} ahz r`nm` a " (p. n} ovufi |h ozh hz lho}k| opn}| fzo mfiiol mzn|nmfi shnk|}+

@ozo n| }`huil fi}h jo kh|ol |`f| mzn|nmfi shnk|} fzo |`o nk|oznhz shnk|} ha fk nk|ozqfi+

(qnn. \`o }|f|nhkfz~ shnk|} fzo |`o shnk|} r`ozo a " (p. 4 3 nk |`o lhdfnk+

Na a (p. n} mhk|nkuhu}) |`ok a (p. mfk bh azhd sh}n|nqo qfiuo} |h kobf|nqo qfiuo} hz azhd

kobf|nqo qfiuo} |h sh}n|nqo hki~ j~ bhnkb |`zhub` |`o qfiuo ozh+ \`o qfiuo ha p ahz r`nm`

a (p. 4 3 fzo fi}h mfiiol |`o |uzknkb shnk|}+ F| f }|f|nhkfz~ hz f |uzknkb shnk|) |`o |fkbok|

|h |`o muzqo ~ 4 a (p. n} sfzfiioi |h |`o p$fpn}+ Hk |`o ioa| fkl znb`| ha f |uzknkb shnk|)

|fkbok|} |h |`o muzqo `fqo lnaaozok| lnzom|nhk+

(qnnn. F aukm|nhk a (p. n} }fnl jo |h nkmzof}nkb hz lomzof}nkb f| f shnk| p3) na |`ozo opn}|} fk

nk|ozqfi (p3 `) p3 ' `.) ` ; 3 fzhukl p3) nk r`nm` |`o aukm|nhk n} nkmzof}nkb hz lomzof}nkb

zo}som|nqoi~+

8/4/2019 wikimama Class 12 Ch 06 Application of Derivatives

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