---
title: "A particle is constrained to move along the \\(x\\)-axis in a region where its potential energy \\(U(x)\\) is given by the smooth function shown in the graph. The curve has a local minimum at position \\(x_1\\), a local maximum at position \\(x_2\\), and an extended horizontal plateau of zero slope and zero curvature at position \\(x_3\\). Which of the following correctly classifies the type of equilibrium at positions \\(x_1\\), \\(x_2\\), and \\(x_3\\)?"
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url: "https://nerd-notes.com/ubq/124009/"
date_modified: "2026-09-28T13:30:11+00:00"
---

# A particle is constrained to move along the \(x\)-axis in a region where its potential energy \(U(x)\) is given by the smooth function shown in the graph. The curve has a local minimum at position \(x_1\), a local maximum at position \(x_2\), and an extended horizontal plateau of zero slope and zero curvature at position \(x_3\). Which of the following correctly classifies the type of equilibrium at positions \(x_1\), \(x_2\), and \(x_3\)?

A particle is constrained to move along the \(x\)-axis in a region where its potential energy \(U(x)\) is given by the smooth function shown in the graph. The curve has a local minimum at position \(x_1\), a local maximum at position \(x_2\), and an extended horizontal plateau of zero slope and zero curvature at position \(x_3\). Which of the following correctly classifies the type of equilibrium at positions \(x_1\), \(x_2\), and \(x_3\)?

![A Cartesian coordinate system with a horizontal axis labeled x and a vertical axis labeled U(x). A single solid curve represents the potential energy function. Starting from the left, the curve dips to a local minimum at a point directly above the horizontal coordinate marked x_1, then rises to a local maximum at a point directly above the horizontal coordinate marked x_2, descends into a flat horizontal plateau with zero slope extending through the horizontal coordinate marked x_3, and finally curves gently upward at the far right. Three vertical dashed lines drop from each of the three marked features on the curve to the horizontal axis at tick marks labeled x_1, x_2, and x_3, respectively. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602210-TbIRVI.jpg)

- **A.** \(x_1\): Unstable | \(x_2\): Stable | \(x_3\): Neutral
- **B.** \(x_1\): Stable | \(x_2\): Unstable | \(x_3\): Neutral
- **C.** \(x_1\): Stable | \(x_2\): Neutral | \(x_3\): Unstable
- **D.** \(x_1\): Neutral | \(x_2\): Unstable | \(x_3\): Stable

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124009/*