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Q.
If the curves $f( x )= e ^{ x }$ and $g ( x )= kx ^2$ touches each other then the value of $k$ is equal to
Application of Derivatives
Solution:
VaodsC Clearly $k>0$
if the point of tangency is $x = x _0$ then $e ^{ x _0}= kx _0^2$ ....(1)
now $f ^{\prime}\left( x _0\right)= g ^{\prime}\left( x _0\right)$
$e ^{ x _0=2 kx _0}$
from (1) $kx _0^2=2 kx _0$
$\therefore x _0=0 \text { and } k =0 \text { is not possible (think!) } $
$ x _0=2$
$\therefore e ^2=4 k \Rightarrow k =\frac{ e ^2}{4}$