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`(log x)/x` का उच्चिष्ठ मान ज्ञात कीजिये यदि ` 0 lt x lt infty ` |
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Answer» माना ` y = (log x)/x ` ...(1) ` :. (dy)/(dx) = (xd/(dx) log x - log x d/(dx) x)/x^(2)` ` rArr (dy)/(dx) = (x *1/x - log x)/x^(2) rArr (dy)/(dx) = (1-log x)/x^(2)` ....(2) तथा ` (d^(2)y)/(dx^(2)) = (x^(2)d/(dx) (1-log x)-(1-log x)d/(dx) x^(2))/x^(4) ` ` = (x^(2)(-1/x) - 2x(1 - log x))/x^(4) = - [(3 -2 log x)/x^(3) ] ` ....(3) फलक के उच्चिष्ठ अथवा निम्निष्ठ मान के लिये ` (dy)/(dx) = 0 rArr (1 - log x)/x^(2) = 0 rArr log x = 1 rArr x = e` ` x = e ` पर , ` (d^(2)y)/(dx^(2)) = - [(3-2 log _(e) e)/e^(3)] = (-1)/e^(3) ` ( ऋण राशि ) अतः x = e पर फलक उच्चिष्ठ है । फलक का उच्चिष्ठ मान , समीकरण (1) से ` y = (log_(e)e)/e = 1/e` ` :. ` फलक का उच्चिष्ठ मान ` 1/e`है । |
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