Question

Find all functions f(x) that have the property that the tangent lines to the graphs of f(x)pass through the point (x+2,0).

Answer 1

Answer:

$y=Ae^{-2x}$

Step-by-step explanation:

Given that the functions f(x) that have the property that the tangent lines to the graphs of f(x)pass through the point (x+2,0).

Let (x,y) be any arbitrary point of contract

The tangent line passes through two points (x,y) and (x+2,0)

Slope of tangent line = f'(x) = change in y/change in x= $\frac{-y}{2}$

i.e. we have

$\frac{dy}{dx} =\frac{-y}{2}$

Separate the variables

$\frac{dy}{y} =-2x\\lny =-2x+c$

Raise to power e

$y=Ae^{-2x}$

Thus the functions would have the above form for various values of A