Question

given the following piece - wise graph, determine the relative maxima and minima.

Explanation:

Step1: Recall definition of relative maxima

A relative (local) maximum is a point where the function changes from increasing to decreasing.

Step2: Analyze the graph

Looking at the graph, at $x = - 4$, the function changes from being constant (horizontal - increasing is not applicable here as it's constant before) to decreasing and $y = 5$. Also at $x=5$, the function changes from increasing to decreasing and $y = 2$.

Step3: Recall definition of relative minima

A relative (local) minimum is a point where the function changes from decreasing to increasing.

Step4: Analyze the graph for minima

At $x=-2$, the function changes from decreasing to increasing and $y = 1$. At $x = 0$, the function is constant on both sides but we can consider it as a local minimum as the value of the function is lower than its surrounding non - constant parts. Also at $x=3$, the function is constant on both sides and can be considered a local minimum.

Answer:

Relative (local) Maxima: $(-4,5),(5,2)$
Relative (local) Minima: $(-2,1),(0, - 1),(3,-1)$