1. **Problem Statement:** Consider a mechanical system where the displacement $x$ of a component depends on the applied force $F$ according to the equation $$F = \frac{k}{x} + cx,$$ where $k$ and $c$ are positive constants representing system parameters. 2. **Goal:** Find the relationship between $F$ and $x$ and describe the shape of the curve when plotted. 3. **Formula and Explanation:** The force equation combines an inverse proportional term $\frac{k}{x}$ and a linear term $cx$. This type of equation often arises in systems with both elastic and inverse-distance dependent forces. 4. **Intermediate Work:** Rewrite the equation as $$F = \frac{k}{x} + cx.$$ To analyze the curve, consider $F$ as a function of $x$: $$F(x) = \frac{k}{x} + cx.$$ 5. **Behavior Analysis:** - For $x > 0$, as $x$ increases, $\frac{k}{x}$ decreases and $cx$ increases. - The sum of these terms creates a curve that decreases sharply for small $x$ and increases linearly for large $x$. 6. **Curve Shape:** This function $F(x)$ forms an upward-opening hyperbolic curve because the dominant term for small $x$ is $\frac{k}{x}$ (hyperbolic decay), and for large $x$ is $cx$ (linear growth). 7. **Final Answer:** The force-displacement relationship is $$F = \frac{k}{x} + cx,$$ which describes an upward-opening hyperbolic curve when graphed.