Question

Analyze the function ln x=cx^2 to find the unique value of c such that there is exactly one solution to the equation. To do this find the value of c such that both sides of the equation have equivalent slopes at some point; this will give you a proper x-coordinate to work with.

Answer 1

following their suggested procedure, ...

slope of lnx = 1/x
slope of xc^2 = 2cx

1/x = 2cx
c = 1/(2x^2)

then lnx = (1/(2x^2))x^2 = 1/2
x = e^(1/2)
x = √e