MathNet 2012 Question 51

Let $f: (0, +\infty) \to (0, +\infty)$ be a twice differentiable function such that $f'(x) \lt 0$ for all $x \gt 0$. For each $x \gt 0$ consider the triangle determined by the line tangent to the graph of $f$ at $(x, f(x))$ and the two coordinate axes. It is known that the area of this triangle is constant and equal to $C$.
Find, in terms of $C$, all possible values of $f(1)$.