Question

identify each characteristic for the graph of the function shown.
decide whether the function has extrema:
extremum relative minimum relative maximum absolute minimum absolute maximum
add extremum no extrema
select a statement that describes the function: both increasing and decreasing

Explanation:

Step1: Analyze the graph's extrema

The graph of the polynomial function has a low point (relative minimum) and a high point (relative maximum), and also, since it's a polynomial with a leading positive coefficient (ends going up), there's no absolute maximum, but let's check the extrema types. First, for "Extremum" type, we can have relative minimum, relative maximum, etc. The graph has a relative minimum (the lowest point) and a relative maximum (the peak), and also, looking at the y - values, the lowest point is a relative minimum, and is there an absolute minimum? Wait, the graph goes down to a low point and then up, but since it's a polynomial, if the degree is even with positive leading coefficient, but here the graph has a "U" - like part and a peak. Wait, the key is: the function has extrema (relative min, relative max, maybe absolute min? Wait, the graph shows a function that has a relative minimum (the bottom of the left - most "U") and a relative maximum (the peak), and also a local minimum (the right - most low point). But the first part: "Extremum" dropdown, we can choose relative minimum, relative maximum, etc. But the question is about whether it has extrema. The graph clearly has peaks and valleys, so it has extrema. Then, for the types:

First, let's identify the extrema. The function has a relative minimum (the lowest point on the left, and another on the right? Wait, the graph: left side goes down to a low point, then up to a peak, then down to a low point, then up. So it has relative minima and a relative maximum. So the function has extrema (so "Add extremum" is selected, not "No extrema").

For the "Extremum" type, let's take relative minimum first? Wait, the first dropdown: "Extremum" can be relative minimum, relative maximum, absolute minimum, absolute maximum. Let's see the graph: the lowest point (the left - most low) is a relative minimum, and the peak is a relative maximum, and the right - most low is also a relative minimum. But the first step: decide whether the function has extrema (it does, so "Add extremum" is correct). Then, for the type of extremum, let's say relative minimum (but maybe the first one is relative minimum).

But the main task here is to identify the characteristics. The function has extrema (so it's not "No extrema"). So we can add extrema. Let's proceed:

Step2: Determine the statement about the function

The function is both increasing and decreasing because it has intervals where it's increasing (e.g., from the relative minimum to the relative maximum, and from the right - most relative minimum to infinity) and intervals where it's decreasing (e.g., from negative infinity to the first relative minimum, and from the relative maximum to the right - most relative minimum). So the statement "Both increasing and decreasing" is correct.

But for the first part: "Decide whether the function has extrema": it does, so we select "Add extremum" and then choose the type. Let's assume we are to fill the "Extremum" dropdown. Let's say the first extremum is a relative minimum.

But maybe the question is about identifying the extrema. Let's focus on the first part: the function has extrema (so "Add extremum" is selected, not "No extrema").

Answer:

The function has extrema (so "Add extremum" is the correct choice for the radio button), and the statement describing the function is "Both increasing and decreasing". For the "Extremum" type, possible choices include relative minimum, relative maximum, etc. If we take the relative minimum as the first extremum, the function has a relative minimum (and other extrema).