Question

1. example 5 p. 244. for each graph, find the absolute maxima and minima. find any local maxima or minima.

Explanation:

Step1: Define absolute and local extrema

Absolute maxima/minima are the highest/lowest points over the entire domain shown. Local maxima/minima are the highest/lowest points in a small - neighborhood.

Step2: Analyze graph (a)

• Absolute max: The highest point is $(3,6)$, so absolute max is $y = 6$ at $x = 3$.
• Absolute min: The lowest point is $(0,1)$, so absolute min is $y = 1$ at $x = 0$.
• Local max: $(3,6)$ (since the function changes from increasing to decreasing).
• Local min: $(4,4)$ (since the function changes from decreasing to increasing).

Step3: Analyze graph (b)

• Absolute max: The highest point in the domain shown is $(5,3)$, so absolute max is $y = 3$ at $x = 5$.
• Absolute min: The lowest point is $(3,1)$ (open - circle is not included in the domain for the min value in this case), so absolute min is $y = 1$ at $x = 3$. There are no local extrema other than the absolute ones as the function is monotonic in sub - intervals.

Step4: Analyze graph (c)

• Absolute max: The highest point is $(5,4)$, so absolute max is $y = 4$ at $x = 5$.
• Absolute min: The lowest points are $(1,1)$ and $(2,1)$, so absolute min is $y = 1$ at $x = 1$ and $x = 2$. There are no local extrema other than the absolute ones as the function is monotonic in sub - intervals.

Step5: Analyze graph (d)

• There is no absolute or local maximum as the function is increasing without bound in the domain shown. The absolute and local minimum is at $(0,0)$ since it is the lowest point and there are no other lower - value points in any neighborhood.

Step6: Analyze graph (e)

• Local max: $(4,3)$ as the function changes from increasing to decreasing around $x = 4$. There is no absolute maximum as the function goes downwards at the right - hand side. The absolute and local minimum does not exist in the strict sense as the points $(1,4)$ and $(2,2)$ are open - circles.

Answer:

Graph (a): Absolute max: $(3,6)$; Absolute min: $(0,1)$; Local max: $(3,6)$; Local min: $(4,4)$
Graph (b): Absolute max: $(5,3)$; Absolute min: $(3,1)$; No non - absolute local extrema
Graph (c): Absolute max: $(5,4)$; Absolute min: $(1,1),(2,1)$; No non - absolute local extrema
Graph (d): No absolute or local max; Absolute and local min: $(0,0)$
Graph (e): Local max: $(4,3)$; No absolute max or min in the strict sense (open - circles for potential min points)