1. ## Tangent lines

I have a problem that I do not know how to start. If anyone could give me a hint on how to start it, that would be great.

Let $\displaystyle f$ be the function given by $\displaystyle f(x) = e^{-x}$, and let $\displaystyle g$ be the function given by $\displaystyle g(x) = kx$, where $\displaystyle k$ is the nonzero constant such that the graph of $\displaystyle f$ is tangent to the graph of $\displaystyle g$.

Find the x-coordinate of the point of tangency and the value of $\displaystyle k$.

2. By condition of tangance df/dx x dg/dx = -1

so$\displaystyle -e^{-x} * k = -1$
$\displaystyle e^{-x} = 1/k$

$\displaystyle e^x = k$
$\displaystyle x = \ln k$

So g(x) is tangent to f(x) at point $\displaystyle x = \ln k$

Now for x = \ln k
you cordinates of both curve should be same

so x = \ln k satisfies
$\displaystyle e^{-x}= kx$

solve for k

3. ## Tangent

Hello Cursed
Originally Posted by Cursed
I have a problem that I do not know how to start. If anyone could give me a hint on how to start it, that would be great.

Let $\displaystyle f$ be the function given by $\displaystyle f(x) = e^{-x}$, and let $\displaystyle g$ be the function given by $\displaystyle g(x) = kx$, where $\displaystyle k$ is the nonzero constant such that the graph of $\displaystyle f$ is tangent to the graph of $\displaystyle g$.

Find the x-coordinate of the point of tangency and the value of $\displaystyle k$.
First, you need to find the value of $\displaystyle x$ where the two graphs meet: that is where $\displaystyle f(x) = g(x)$, or

$\displaystyle e^{-x}= kx$ (1)

Then if $\displaystyle g(x)$ is a tangent to $\displaystyle f(x)$ for this value of $\displaystyle x$, their gradients are equal at this point. So $\displaystyle f'(x) = g'(x)$ or:

$\displaystyle -e^{-x} = k$

$\displaystyle \Rightarrow e^{-x} = -k$ (2)

Now you can use (2) to eliminate the $\displaystyle e^{-x}$ term in (1), and solve the resulting equation for $\displaystyle x$. This is the $\displaystyle x$-coordinate of the point of tangency, which you can then use to find the value of $\displaystyle k$, by substituting it back into (1).

Can you complete it now?