---
title: "A small automated cart moves along a straight, horizontal track aligned with the x-axis. The graph shows the cart’s position \\(x\\) as a function of time \\(t\\). At time \\(t = t_0\\), the position-time curve has an inflection point, transitioning from concave downward to concave upward, while the tangent line to the curve at \\(t = t_0\\) has a positive slope.  Which of the following statements correctly describes the cart’s motion at or near time \\(t = t_0\\)?"
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url: "https://nerd-notes.com/ubq/123970/"
date_modified: "2026-09-28T13:29:42+00:00"
---

# A small automated cart moves along a straight, horizontal track aligned with the x-axis. The graph shows the cart’s position \(x\) as a function of time \(t\). At time \(t = t_0\), the position-time curve has an inflection point, transitioning from concave downward to concave upward, while the tangent line to the curve at \(t = t_0\) has a positive slope.

Which of the following statements correctly describes the cart’s motion at or near time \(t = t_0\)?

A small automated cart moves along a straight, horizontal track aligned with the x-axis. The graph shows the cart's position \(x\) as a function of time \(t\). At time \(t = t_0\), the position-time curve has an inflection point, transitioning from concave downward to concave upward, while the tangent line to the curve at \(t = t_0\) has a positive slope.

Which of the following statements correctly describes the cart's motion at or near time \(t = t_0\)?

![A graph on bare Cartesian axes. The horizontal axis is labeled t, and the vertical axis is labeled x, each with an arrowhead at its positive end. The origin is labeled O. A single smooth, solid curve begins near O and increases monotonically into the first quadrant. For t < t_0, the curve is concave downward with a decreasing positive slope. At a specific time labeled t_0 on the horizontal axis, marked by a vertical dashed line connecting the axis to the curve, the curve passes through an inflection point with a positive slope. For t > t_0, the curve is concave upward with an increasing positive slope. No other labels, lines, text, or tick marks appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602182-1d9Eq7.jpg)

- **A.** At \(t = t_0\), the instantaneous acceleration of the cart is zero, and its speed is at a local minimum.
- **B.** At \(t = t_0\), the instantaneous acceleration of the cart is at a local maximum, and its speed is zero.
- **C.** At \(t = t_0\), the instantaneous acceleration of the cart is zero, and its speed is at a local maximum.
- **D.** At \(t = t_0\), the instantaneous velocity of the cart is zero, and its acceleration changes from positive to negative.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123970/*