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(B) The curves $y=e^x$ and $y=\log_e x$ are inverse functions of each other,meaning they are symmetric about the line $y=x$.Let $A$ be a point on $y=e

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$\sqrt{2}$

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(B) The curves $y=e^x$ and $y=\log_e x$ are inverse functions of each other,meaning they are symmetric about the line $y=x$.Let $A$ be a point on $y=e^x$ with coordinates $(h, e^h)$. The distance from point $A$ to the line $y=x$ (or $x-y=0$) is given by $AB = \frac{|h-e^h|}{\sqrt{1^2+(-1)^2}} = \frac{|h-e^h|}{\sqrt{2}}$.Since the curves are symmetric about $y=x$,the minimum distance between the two curves is $AC = 2AB$,where $B$ is the projection of $A$ onto the line $y=x$.To minimize $AB$,we minimize $f(h) = e^h - h$ (since $e^h > h$ for all $h$).$f'(h) = e^h - 1$. Setting $f'(h) = 0$ gives $e^h = 1$,so $h=0$.At $h=0$,the minimum distance $AB = \frac{|0-e^0|}{\sqrt{2}} = \frac{1}{\sqrt{2}}$.Therefore,the minimum distance between the curves is $AC = 2 	imes \frac{1}{\sqrt{2}} = \sqrt{2}$.

The minimum distance between a point on the curve $y=e^x$ and a point on the curve $y=\log_e x$ is

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