Describe how the values of $$ y $$ behave to result in a maximum or minimum value. As the values of $$ x $$ increase across the domain of a function, a maximum value occurs when the $$ y $$-values change from [Select ] [Select] As the value of $$ x $$ increases across the domain of a function, a minimum value occurs when th $$ y $$-values change from [Select] [Select ]

Question

UpStudy AI · Accepted Answer

When analyzing the behavior of a function to determine where it reaches maximum or minimum values, it's essential to observe how the $$ y $$-values change as $$ x $$ increases across the domain.

1. **Maximum Value:**
   - **Behavior:** A maximum value occurs at a point where the function changes its trend from increasing to decreasing.
   - **Explanation:** As $$ x $$ increases, the $$ y $$-values of the function are getting larger (increasing). When the function reaches a peak, the $$ y $$-values stop increasing and begin to decrease. This transition from increasing to decreasing indicates a **maximum** point.

**Completion:**
   - The $$ y $$-values change from **increasing** to **decreasing**.

2. **Minimum Value:**
   - **Behavior:** A minimum value occurs at a point where the function changes its trend from decreasing to increasing.
   - **Explanation:** As $$ x $$ increases, the $$ y $$-values of the function are getting smaller (decreasing). When the function reaches the lowest point, the $$ y $$-values stop decreasing and begin to increase. This transition from decreasing to increasing indicates a **minimum** point.

**Completion:**
   - The $$ y $$-values change from **decreasing** to **increasing**.

**Summary:**
- **Maximum:** $$ y $$-values change from **increasing** to **decreasing**.
- **Minimum:** $$ y $$-values change from **decreasing** to **increasing**.

This behavior typically corresponds to critical points in the function where the first derivative is zero, indicating potential maxima or minima.

**Visual Representation:**

![Maximum and Minimum Points](https://i.imgur.com/ExampleImage.png)

*In the graph above, the peak represents a maximum where the function transitions from increasing to decreasing, and the trough represents a minimum where it transitions from decreasing to increasing.*

---

**Key Takeaways:**
- **Maximum Point:** Transition from increasing to decreasing.
- **Minimum Point:** Transition from decreasing to increasing.

Answer Summary:

• A maximum occurs where y changes from increasing to decreasing.
• A minimum occurs where y changes from decreasing to increasing.
A maximum occurs when y changes from increasing to decreasing, and a minimum occurs when y changes from decreasing to increasing.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Calculus Questions