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Q.
The value of $k$ for which the equation $e ^{2 x }= k \sqrt{ x }$ has exactly one solution, is
Application of Derivatives
Solution:
$ k \text { can not be negative } $
$\frac{ dy }{ dx }|_{ P }=2 e ^{2 x _1} $
$\therefore \frac{ dy }{ dx }|_{ P }=\frac{ k }{2 \sqrt{ x _{ l }}} $
$2 e ^{2 x _1}=\frac{ k }{2 \sqrt{ x _1}}\left(\text { but } e ^{2 x _1}= k \sqrt{ x _1}\right) $
$2 k \sqrt{ x _1}=\frac{ k }{2 \sqrt{ x _1}} ; x _1=\frac{1}{4} ; $
$ \therefore e ^{\frac{1}{2}}=\frac{ k }{2} ; k =2 \sqrt{ e } $
