Hello gwtkof
Welcome to Math Help Forum! Originally Posted by
gwtkof y=cos(x) intersects y=x once,
find A such that y=Acos(x) intersects y=x exactly twice.
I think you will need to use a numerical/graphical method to solve this.
The second point of intersection will be when the line is a tangent to the curve between $\displaystyle x=0$ and $\displaystyle x=-\pi$.
Using an Excel spreadsheet, I set up graphs of the two functions, and found, by a series of approximations, this to be when $\displaystyle x \approx -2.8$, with $\displaystyle A\approx 2.972$.
Grandad
PS
A more analytical approach is to say that the gradient of $\displaystyle A\cos x$ at the point when it touches the line will be $\displaystyle \displaystyle 1$. So if you solve the equations
$\displaystyle A\cos x=x$ and $\displaystyle -A\sin x = 1$
simultaneously, you'll end up with
$\displaystyle \tan x = -\dfrac1x$
which will again require a numerical method.
This gives the same approximate values as I stated above.