---
title: "A single conservative force \\(F_x\\) acts on a particle constrained to move along the \\(x\\)-axis. The graph of \\(F_x\\) as a function of position \\(x\\) is shown, where the curve crosses the horizontal axis at \\(x = x_1\\) and \\(x = x_3\\), and reaches a local maximum at \\(x = x_2\\).  Which of the following statements correctly identifies the location of a local minimum in the particle’s potential energy \\(U(x)\\) and provides a valid justification?"
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url: "https://nerd-notes.com/ubq/120640/"
date_modified: "2026-08-23T04:42:27+00:00"
---

# A single conservative force \(F_x\) acts on a particle constrained to move along the \(x\)-axis. The graph of \(F_x\) as a function of position \(x\) is shown, where the curve crosses the horizontal axis at \(x = x_1\) and \(x = x_3\), and reaches a local maximum at \(x = x_2\).

Which of the following statements correctly identifies the location of a local minimum in the particle’s potential energy \(U(x)\) and provides a valid justification?

A single conservative force \(F_x\) acts on a particle constrained to move along the \(x\)-axis. The graph of \(F_x\) as a function of position \(x\) is shown, where the curve crosses the horizontal axis at \(x = x_1\) and \(x = x_3\), and reaches a local maximum at \(x = x_2\).

Which of the following statements correctly identifies the location of a local minimum in the particle's potential energy \(U(x)\) and provides a valid justification?

![A Cartesian coordinate graph with a horizontal axis labeled x and a vertical axis labeled F_x. A horizontal dashed axis line represents F_x = 0. A smooth continuous solid curve begins in the negative F_x region to the left, rises upward to cross the horizontal axis at a point marked x_1 with a positive slope, reaches a peak in the positive F_x region marked x_2, and then slopes downward to cross the horizontal axis at a point marked x_3 with a negative slope before continuing downward into the negative F_x region. Vertical tick marks labeled x_1, x_2, and x_3 sit along the horizontal axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460147-9vAIe1.jpg)

- **A.** The potential energy has a local minimum at \(x = x_1\), because the net force is zero and the slope \(\dfrac{dF_x}{dx}\) is positive.
- **B.** The potential energy has a local minimum at \(x = x_2\), because the derivative \(\dfrac{dF_x}{dx} = 0\) at that point.
- **C.** The potential energy has a local minimum at \(x = x_3\), because \(F_x = 0\) and the negative slope \(\dfrac{dF_x}{dx} < 0\) corresponds to \(\dfrac{d^2U}{dx^2} > 0\).
- **D.** The potential energy has a local minimum at \(x = x_3\), because the total area under the curve between \(x_1\) and \(x_3\) is positive.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120640/*