1. **Problem Statement:** We are given a function $y = g(x)$ with information about its absolute and local extrema. 2. **Definitions:** - An **absolute maximum** is the highest point on the entire graph. - An **absolute minimum** is the lowest point on the entire graph. - A **local maximum** is a point where the function value is higher than all nearby points. - A **local minimum** is a point where the function value is lower than all nearby points. 3. **Given Data:** - Absolute maximum at $x=4$ - Absolute minimum at $x=1$ 4. **Local Extrema Analysis:** - The graph has a local maximum between $x=1$ and $x=2$. - There is a local minimum around $x=3$. - There is a less tall local maximum at $x=4$ (which is also the absolute maximum). 5. **Summary of Extrema:** - Absolute maximum: $x=4$ - Absolute minimum: $x=1$ - Local maxima: one between $x=1$ and $x=2$, and one at $x=4$ - Local minimum: around $x=3$ 6. **Explanation:** - The absolute maximum at $x=4$ means $g(4)$ is the highest value on the graph. - The absolute minimum at $x=1$ means $g(1)$ is the lowest value on the graph. - Local maxima and minima are points where the slope changes from positive to negative or vice versa, indicating peaks and valleys. This analysis helps understand the behavior of $g(x)$ based on the graph description and extrema points.