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E L-LOVELY-H math9-1 IInd 6-8-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12 VIth 10-9-12

noEPk;soh dk rfDs

B'N wzfBnk fpzd{ x = a d/ yZp/ gk;/ (L.H.S.) ;fEs e'Jh B/Vbk fpzd{ x = a – h ns/ x = a d/ ;Zi/ gk;/
(R.H.S.) ;fEs e'Jh B/Vbk fpzd{ x = a + h j?, fiZE/ h pj[s fInkdk nbg okPh j?.
fpzd{ x = a T[Zs/ cbB f (x) dk wkB = f (a)
fpzd{ x = (a – h) T[Zs/ cbB f (x) dk wkB = f (a – h)
ns/ fpzd{ x = (a + h) T[Zs/ cbB f (x) dk wkB = f (a + h)H

fpzd{ x = a T[Zs/ cbB f (x) T[uso ejkT[Adk j?, i/eo
f (a – h) < f (a) > f (a + h)

Gkt x = a T[Zs/ f (x) dk wkB f (a), yZp/ ns/ ;Zi/ ;fEs B/Vb/ fpzd{nK T[Zs/ T[;d/ wkBK f (a – h)
ns/ f (a + h) d'jK s'A tZvk j?.

B'N; e'Jh cbB f (x), x = a T[Zs/ T[d'A T[uso (maximum) ejkT[Adk j?, id'A f (a) T[jBK
;ko/ wkBK s'A tZvk j't/ fijVk x = a d/ bx{ B/Vb/ ftZu x jo/e wkB d/ bJh f (x)
rqfjD eo ;edk j?.

fBwBso (Minima)
wzB fbU teo T[Zs/ ;fEs fpzd{ B d/ bJh x = a = OM
∴ x = a T[Zs/, y = f (a) = BM.
j[D fpzd{ B d/ yZp/ gk;/ bx{ B/Vb/ (small neighbourhood) ftZu e'Jh fpzd{ B 1 fbU fi;d/ bJh
x = a – h = OM 1 , fiZE/ h pj[s S'Nk j?.
fJ; soQK x = a – h T[Zs/, y = f (a – h) = B 1 M 1 > BM.
fJ;h soQK fpzd{ B d/ ;Zi/ gk;/ bx{ B/Vb/ (small neighbourhood) ftZu e'Jh fpzd{ B 2 fbU fi;d/
bJh x = a + h = OM 2 .
fJ; soQK x = a + h T[Zs/, y = f (a + h) = B 2 M 2 > BM.
j[D feT[Ai' B 1 M 1 , BM s'A tZvk j?, Gkt B 1 M 1 > BM

f (a – h) > f (a) iK f (a) < f (a – h)
ns/ feT[Ai' B 2 M 2 , BM s'A tZvk j?, Gkt B 2 M 2 > BM.
f (a + h) > f (a) iK f (a) < f (a + h)
fJ; soQK fpzd{ B (fi;d/ bJh x = a) T[Zs/ f (x) dk wkB, Gkt f (a), B d/ yZp/ iK ;Zi/ jo fpzd{
T[Zs/ f (x) d/ ;zrs wkBK, Gkt f (a – h) ns/ f (a + h) s'A S'Nk j?.
fpzd{ x = a T[Zs/ cbB f (x) fBwBso ejkT[Adk j?, i/eo
f (a – h) > f (a) < f (a + h)

Gkt x = a s'A f (x) dk wkB f (a), yZp/ ns/ ;Zi/ ;fEs B/Vb/ fpzd{nK T[Zs/ T[;d/ wkBK f (a – h) ns/
f (a + h) d'jK s'A S'Nk j?.
fe;h cbB d/ T[uso wkB (Maximum) dk fJj noE BjhA j? fe T[j T[[;dk ;G s'A tZvk wkB j?
ns/ Bk fBwBso wkB dk fJj noE j? fe T[j ;G s'A S'Nk wkB j?. fe;h cbB d/ pj[s ;ko/
T[uso ns/ fBwBso wkB j' ;ed/ jB ns/ fJj ;zGt j? fe T[uso wkB d{i/ fBwBso wkB s'A
S'Nk j't/. A cbB iK e'fN dk T[uso wkB j?, fJ;dk noE fJj j? fe fJ; fpzd{ d/ nk;gk;
tkb/ d'jK fpzd{nK s'A fJ;dk wkB ;G s'A fInkdk j? ns/ fJ;h soQK fpzd{ B T[Zs/ cbB iK e'fN dk

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