The path of a specialized micro-drone as it ascends through a restricted aerodynamic corridor is modeled by a curve C C\,C in the xyxyxy-plane. The horizontal displacement xxx (in decametres) of the drone is related to its vertical height yyy (in decametres) by the equation
x=ye−5y,y∈R x = y e^{-5y}, \quad y \in \mathbb{R} x=ye−5y,y∈RShow that
dydx=yx(1−5y) \frac{dy}{dx} = \frac{y}{x(1 - 5y)} dxdy=x(1−5y)yGiven that the vertical line with equation x=kx = kx=k, where k k\,k is a constant, intersects C C\,C at exactly two points,
find the range of possible values for kkk.
