Question

intervals of polynomials
key strategies:

example: where is the graph increasing? decreasing?

you try it: where is the graph below increasing?

you try it: where is the graph decreasing:

maxima/ minima
example: find all relative maximums and minimums.
state if any is an absolute maximum or minimum.

you try it: find all relative maximums and minimums.
state if any is an absolute maximum or minimum.

relative minimum:
relative maximum:
absolute maximum:
absolute minimum:

relative minimum:
relative maximum:
absolute maximum:
absolute minimum:

Explanation:

Step1: Analyze the first "You try it" graph (increasing)

The graph has a minimum at \( x = 1 \) (y - value - 1), then increases from \( x = 1 \) to \( x = 3 \), has a small decrease, then increases from \( x = 4 \) to \( x = 6 \). Also, from \( x = 0 \) to \( x = 1 \), it's decreasing, but we focus on increasing. So the increasing intervals are where the slope is positive. Looking at the graph: from \( x = 1 \) to \( x = 3 \), then from \( x = 4 \) to \( x = 6 \), and also check the left - most part? Wait, no, the graph starts at \( x = 0 \) with \( y = 4 \), decreases to \( x = 1 \), then increases. Wait, the x - axis: the graph is plotted from \( x = 0 \) to \( x = 6 \). Let's list the intervals where the function's y - value increases as x increases. At \( x = 1 \), it's a minimum. Then from \( x = 1 \) to \( x = 3 \), y increases (from - 1 to 1). Then from \( x = 4 \) to \( x = 6 \), y increases (from the minimum at \( x = 4 \) to a high value at \( x = 6 \)). Also, wait, the left - hand side: from \( x = 0 \) to \( x = 1 \), it's decreasing (y goes from 4 to - 1). So the increasing intervals are \( (1, 3) \) and \( (4, 6) \)? Wait, no, let's check the coordinates. The point at \( x = 1 \) is \( (1, - 1) \), \( x = 3 \) is \( (3, 1) \), \( x = 4 \) is \( (4, 0.5) \) (approx), and \( x = 6 \) is \( (6, 4) \). So the function increases when moving from \( x = 1 \) to \( x = 3 \) (slope positive), then decreases from \( x = 3 \) to \( x = 4 \) (slope negative), then increases from \( x = 4 \) to \( x = 6 \) (slope positive). Also, is there an interval before \( x = 1 \)? The graph starts at \( x = 0 \), \( y = 4 \), and goes down to \( x = 1 \), \( y=-1 \), so that's decreasing. So the increasing intervals are \( (1, 3) \) and \( (4, 6) \).

Step2: Analyze the second "You try it" graph (decreasing)

This graph has multiple segments. Let's identify the intervals where the y - value decreases as x increases. First, the left - most segment: from the left end to the minimum point, then after some segments, find where the slope is negative. Let's look at the key points. The graph has a minimum, then increases, then a horizontal segment, then increases, then decreases, then increases. Wait, the x - axis: let's assume the key points. The first minimum: then it increases, then a horizontal line (slope 0, not decreasing), then increases to a peak, then decreases to a minimum, then increases. So the decreasing intervals are: from the peak to the next minimum, and the initial left - most segment? Wait, the left - most part: from the left end to the first minimum (slope negative, so decreasing). Then from the first peak (after the horizontal segment) to the next minimum (slope negative, decreasing). Let's denote the x - values. Suppose the first minimum is at \( x = a \), then increases to \( x = b \), horizontal from \( x = b \) to \( x = c \), then increases to \( x = d \) (peak), then decreases to \( x = e \) (minimum), then increases. So the decreasing intervals are \( (-\infty, a) \) (left - most to first minimum), \( (d, e) \) (peak to next minimum).

Step3: Analyze the Maxima/Minima "You try it" graph

For the relative maxima and minima, we look for points where the function changes from increasing to decreasing (relative maximum) or decreasing to increasing (relative minimum). The graph has a peak (relative maximum) at \( x=-3 \) (from the example - like graph), then a minimum at the bottom of the curve, then a peak at \( x = 4 \)? Wait, no, the graph is similar to the example. The relative maximum is at the top of the left - hand curve (around \( x=-3 \)), then a relative minimum at the bottom of the middle curve (around \( x = 1 \) - 2? Wait, the example graph has a relative maximum at \( x=-3 \), then a relative minimum at the bottom of the curve (around \( x = 1 \) - 2), then a relative maximum at \( x = 4 \)? No, the example graph: the left - hand curve goes up to a peak at \( x=-3 \), then down to a minimum at \( x = 1 \) - 2, then up to \( x = 4 \). So relative maximum at \( x=-3 \) (since the function changes from increasing to decreasing there), relative minimum at the bottom of the curve (function changes from decreasing to increasing there). Absolute maximum: if the graph goes to infinity, but in the example, the graph has a peak at \( x=-3 \) and then goes up at \( x = 4 \). Wait, no, the graph in the "You try it" for maxima/minima: the left - hand curve has a minimum, then a maximum, then a minimum, then a maximum? Wait, the first graph for maxima/minima (left) has two minima and one maximum. The relative minima are at the bottom of the left - hand curve and the bottom of the right - hand curve. The relative maximum is in the middle. Absolute maximum: since the graph goes to infinity (the ends go up), there is no absolute maximum. Absolute minimum: the lowest points, but if the graph goes to negative infinity? No, the graph is plotted with a grid, so maybe the relative minima are the local lows, and absolute minimum would be if it's the lowest overall.

But let's focus on the first "You try it" (increasing) graph. Let's re - examine:

The x - values: the graph is on the coordinate plane with x from 0 to 6. The critical points (where slope changes) are at \( x = 1 \), \( x = 3 \), \( x = 4 \).

- From \( x = 0 \) to \( x = 1 \): y decreases (from 4 to - 1) → decreasing.
- From \( x = 1 \) to \( x = 3 \): y increases (from - 1 to 1) → increasing.
- From \( x = 3 \) to \( x = 4 \): y decreases (from 1 to 0.5) → decreasing.
- From \( x = 4 \) to \( x = 6 \): y increases (from 0.5 to 4) → increasing.

So the increasing intervals are \( (1, 3) \) and \( (4, 6) \).

For the decreasing graph (second "You try it"):

Let's assume the graph has the following key points (from left to right):

1. A minimum (let's say at \( x = m_1 \)), then increases to a point, then a horizontal segment (slope 0), then increases to a peak ( \( x = p_1 \) ), then decreases to a minimum ( \( x = m_2 \) ), then increases.

- The interval from the left end to \( x = m_1 \): decreasing (slope negative).
- The interval from \( x = p_1 \) to \( x = m_2 \): decreasing (slope negative).

For the maxima/minima "You try it" graph (similar to the example):

- Relative minimum: the lowest points in their local neighborhoods. In the example - like graph, the bottom of the curve (around \( x = 1 \) - 2) is a relative minimum (function changes from decreasing to increasing there).
- Relative maximum: the top of the left - hand curve (around \( x=-3 \)) is a relative maximum (function changes from increasing to decreasing there).
- Absolute maximum: if the graph has a highest point overall. If the graph goes to infinity, there is no absolute maximum. If not, the highest point.
- Absolute minimum: if the graph has a lowest point overall. If the graph goes to negative infinity, there is no absolute minimum. If not, the lowest point.

Answer:

For the first "You try it" (increasing graph): The graph is increasing on the intervals \( (1, 3) \) and \( (4, 6) \).

For the second "You try it" (decreasing graph): The graph is decreasing on intervals such as (left - most to first minimum) and (peak to next minimum) (exact intervals depend on precise coordinates, but the method is to find where y decreases as x increases).

For the maxima/minima "You try it" graph:

• Relative Minimum: The lowest points in local regions (e.g., the bottom of the middle curve).
• Relative Maximum: The highest points in local regions (e.g., the top of the left - hand curve).
• Absolute Maximum: None (if the graph extends to infinity) or the highest point (if bounded).
• Absolute Minimum: None (if the graph extends to negative infinity) or the lowest point (if bounded).

(Note: Since the problem has multiple "You try it" parts, the above answers address each part with the method of analyzing the graph's slope for increasing/decreasing and critical points for maxima/minima.)