Question

If the curves $f(x)=e^x$ and $g(x)=k{x}^2$ touches each other then the value of $k$ is equal to

• A. 2
• B. $\frac{e^2}{4}$
• C. $\frac{e^2}{2}$
• D. $\frac{e}{2}$

Answer 1

Answer: (B)

VaodsC Clearly $k>0$
if the point of tangency is $x=x_0$ then $e^{x_0}=k{x}_0^2$ ....(1)
now $f^{\prime }\left(x_0\right)=g^{\prime }\left(x_0\right)$
$e^{x_0=2k{x}_0}$
from (1) $k{x}_0^2=2k{x}_0$
$\therefore x_0=0\text{ and }k=0\text{ is not possible (think!) }$
$x_0=2$
$\therefore e^2=4k\Rightarrow k=\frac{e^2}{4}$