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E L-LOVELY-H math9-1 IInd 21-10-11 IIIrd 24-1-12 IVth 21-4-12 Vth 20-8-12 VIth 10-9-12
fJekJh^9 L T[uso ns/ fBwBso L fJZe uo
;fEsh d/ nkT[D ftZu cbB fBwBso wkB gqkgs eodk j?. fJ; soQK ;gPN j? fe T[uso fpzd{ T[Zs/ B'N
cbB dk wkB T[;d/ mhe gfjbK ns/ mhe pknd (small neighbourhood) d/ fpzd{nK T[Zs/ wkBK s'A
tZX j[zdk j? ns/ fBwBso fpzd{ T[Zs/ T[;d/ mhe gfjbK ns/ T[;d/ mhe pknd (small
neighbourhood) d/ fpzd{nK T[Zs/ d/ wkBK s'A xZN j[zdk j?.
3
2
T[dkjoD L wzB fbU cbB y = x – 6x – 2,
x d/ ;ko/ wkBK d/ bJh fijVk 1 s'A xZN j?.
Gkt (x < 1) y dk wkB th tZXdk j? ns/
1 < x < 3 d/ bJh y dk wkB xNdk j?.
fJ; soQK fJ; d/ wZX y dk wkB fJZe T[Zuso
wkB j?.
wzB fbU y = f (x), fJZe cbB j?. y = f (x)
dk nkb/y fuZso d/ nB[;ko fBo{fgs j?.
fJ; teo T[Zs/ A, B, C, D uko fpzd{ jB
fiBQK T[Zs/ ;goP o/yktK x-neP d/ ;wKso
jB.
j[D cbB f (x) dk wkB A ns/ C T[Zs/ fInkdk j? ns/ A ns/ C d/ nZr/ tXD s/ f (x) d/ wkB dk
tXDk o[e iKdk j?, gozs{ xNDk nozG j' iKdk j?. fJ; bJh A ns/ C fpzd{nK T[Zs/ f (x)
fInkdkso wkB rqfjD eodk j?. fJ; soQK fpzd{ A ns/ C T[Zs/ cbB dk wkB T[Zuso (Maximum)
ejkT[Adk j?. fJ;h soQK B ns/ D T[Zs/ cbB dk wkB xNDk pzd j' iKdk j? Gkt nZr/ tXD s/ f (x)
dk tXDk nozG j' iKdk j?. fJ; soQK f (x) cbB B ns/ D T[Zs/ i' wkB gqkgs eodk j? T[jBK B{z
fBwBso (Minimum) efjzd/ jB.
T[uso (Maxima)
wzB fbU teo T[Zs/ ;fEo fpzd{ A d/ bJh x = a = OL
/
∴ x = a T[Zs, y = f (a) = AL
fpzd{ A d/ yZp/ gk;/ bx{ B/Vsk (small neighbourhood) ftZu e'Jh fpzd{ A 1 fbU fi;d/ bJh x = a –
h = OL 1 , fJZE/ h pj[s S'Nk j?.
fJ; soQK x = a – h T[Zs/, y = f (a – h) = A 1 L 1 < AL
fJ;h soQK fpzd{ A d/ ;Zi/ gk;/ bx{ B/Vsk (small neighbourhood) ftZu e'Jh fpzd{ A 2 fbU fi; d/
bJh
x = a + h = OL 2 .
fJ; soQK x = a + h T[Zs/, y = f (a + h) = A 2 L 2 < AL
j[D feT[Ai'A A 1 L 1 , AL s'A S'Nk j?, Gkt A 1 L 1 < AL
∴ f (a – h) < f (a) iK f (a) > f (a – h)
ns/ feT[aAi'A A 2 L 2 , AL s'A S'Nk j?, Gkt A 2 L 2 < AL
∴ f (a +h) < f (a) iK f (a) > f (a + h)
fJ; soQK A (fi;d/ bJh x = a) T[Zs/ f (x) dk wkB, Gkt f (a) A d/ yZp/ iK ;Zi/ jo fpzd{ T[Zs/ f (x)
d/ ;zrs wkBK, Gkt f (a – h) ns/ f (a + h) s'A tZvk j?.
9H2 T[uso ns/ fBwBso dh gfoGkPk (Definition of Maxima and Minima)
T[Zuso (Maxima)
wzB fbU y = f (x) e'Jh fdZsk j'fJnk cbB j? ns/ x = a e'Jh fdZsk j'fJnk fpzd{ j?.
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