Question

directions: for each of the following polynomial graphs, determine any x - values where the graph has a relative extrema. if the graph does not have a specific relative extrema, write “none” in the appropriate space.
relative minimums at x = - 1,4,9
relative maximums at x = - 2,2,8
relative minimums at x = 2,6
relative maximums at x = - 4,5
directions: for each of the following polynomial graphs, determine the absolute minimum and absolute maximum. if the graph does not have a specific absolute extrema, write “none” in the appropriate space.
absolute minimum = 2 at x = 4
absolute maximum = none at x
absolute minimum = none at x
absolute maximum = 8 at x = 7

Explanation:

Step1: Recall definitions

Relative extrema occur where the graph changes from increasing to decreasing (relative maximum) or vice - versa (relative minimum). Absolute extrema are the overall highest and lowest points on the graph.

Step2: Analyze first graph (top - left)

For relative minima, we look for valleys. The graph has relative minima at $x=-1,4,9$. For relative maxima, we look for peaks. The graph has relative maxima at $x = - 2,2,8$.

Step3: Analyze second graph (top - right)

The relative minima are at $x = 2,6$ and the relative maxima are at $x=-4,5$.

Step4: Analyze third graph (bottom - left)

The absolute minimum occurs at the lowest point of the graph. Here, the absolute minimum is $y = 2$ at some $x$ - value (not clearly defined in the image). There is no absolute maximum as the graph extends upwards on both ends.

Step5: Analyze fourth graph (bottom - right)

There is no absolute minimum as the graph extends downwards on the left - hand side. The absolute maximum is $y = 8$ at $x = 7$.

Answer:

For the first graph:
Relative minima at $x=-1,4,9$; Relative maxima at $x=-2,2,8$
For the second graph:
Relative minima at $x = 2,6$; Relative maxima at $x=-4,5$
For the third graph:
Absolute minimum is $2$ (at an unspecified $x$); Absolute maximum is none
For the fourth graph:
Absolute minimum is none; Absolute maximum is $8$ at $x = 7$