If the two curves $\displaystyle y=e^{2x}$ and $\displaystyle y=k\sqrt{x}$ (wherekis a constant) have exactly one point in common, what much be the value ofk?

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- Oct 18th 2007, 12:22 PMpolymeraseFinding a point in common
If the two curves $\displaystyle y=e^{2x}$ and $\displaystyle y=k\sqrt{x}$ (where

*k*is a constant) have exactly one point in common, what much be the value of*k*? - Oct 18th 2007, 02:17 PMangel.white
It looks like

$\displaystyle e^{2x}=k\sqrt{x}$

which means

$\displaystyle \frac{e^{2x}}{\sqrt{x}}=k$

I think you would need to know the point where the two equations intersect to determine what the actual value of k is. - Oct 18th 2007, 07:15 PMpolymerase
- Oct 18th 2007, 08:45 PMJhevon
$\displaystyle k = 2 \sqrt{e}$ is the answer

Hint: Let one of the functions be $\displaystyle y_1$ and the other be $\displaystyle y_2$

we must fulfill two conditions.

(1) we must have $\displaystyle y_1 = y_2$

(2) we must have $\displaystyle y_1' = y_2'$

the first condition is obvious, but can you tell me why we need the second? after that, can you come up with the solution? - Oct 18th 2007, 10:03 PMangel.white
I can't, and I really want to know, it may be a misconception on my part, but when I was thinking about it, I figured two different shaped curves could touch the same point, giving each a different tangent line. I figured if they were circles, they would have the same tangents, but since they weren't, they wouldn't.

If I had to guess at an answer, I'd say it's because they don't cross each other, if they could cross each other they could clearly touch the same point with different tangent lines. But, thats just a guess.

Also, how would you solve for x at this point? I can see that the point would be $\displaystyle (\frac{1}{4}, \sqrt{e})$ But I don't know how to solve for it without just being able to "see" the answer. - Oct 19th 2007, 04:42 AMtopsquark