Question

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

Explanation:

Step1: Recall definition of relative maxima

Relative maxima are points where the function changes from increasing to decreasing.

Step2: Analyze the graph

Looking at the graph, at $x = - 8$, the function has an open - circle at $(-8,5)$ and is constant until $x=-4$. Also, at $x = 5$, the function changes from increasing to decreasing. The $y$ - value at $x=-8$ (open - circle) is not a relative maximum as the function is not defined at that exact point. The relative maximum occurs at the point $(5,2)$.

Step3: Recall definition of relative minima

Relative minima are points where the function changes from decreasing to increasing.

Step4: Analyze the graph for minima

At $x=-2$, the function changes from decreasing to increasing with a $y$ - value of $1$, and at $x = 0$, the function has a relative minimum with a $y$ - value of $-1$.

Answer:

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