Question

cating local maximum and minimum function values use the graph to find the local minimum and the local maximum for the given function. hover over (or tap) a point to see its coordinates. over the interval -3, -1, the local minimum is over the interval -1, 0, the local maximum is over the interval 0, 3, the local minimum is

Explanation:

Step1: Analyze interval [-3, -1]

In the interval \([-3, -1]\), the graph has a point at \(x = -2\) with \(y = 0\) (local minimum as it's the lowest in this interval).

Step2: Analyze interval [-1, 0]

In \([-1, 0]\), the peak (local maximum) is at \(x\) between -1 and 0, with \(y\) value (from graph) around 5? Wait, looking at the graph, the point between -1 and 0 has \(y\) coordinate (let's check the graph: the point near -1 to 0, the y - value is, from the grid, the point at x between -1 and 0, the y is, say, when x is -1 to 0, the local maximum is at ( - 0.5, 5)? Wait no, looking at the graph, the point at x=-1 to 0: the graph has a point at x=-1 to 0, the y - coordinate is, from the graph, the local maximum in [-1,0] is at ( - 1, 5)? Wait no, the graph shows a point at x=-2 (y=0), then a rise to a point (let's say x=-1, y=5? Wait, the graph: at x=-2, y=0; then it goes up to a point (maybe x=-1, y=5) then down to (0,0). Wait, no, the graph: the left part: at x=-3, y is around 20, then down to x=-2, y=0, then up to a point (x=-1, y=5) then down to (0,0). So in [-1,0], the local maximum is at x=-1 (or near) with y=5? Wait, no, the graph's point between -1 and 0: the local maximum in [-1,0] is the highest point in that interval. So the point at x=-1 to 0, the y - value is, let's see the grid. The y - axis has 10, 20, 0, -10, -20, -30. The point between -1 and 0: the y - coordinate is 5? Wait, maybe the local maximum in [-1,0] is 5? Wait, no, looking at the graph, the point at x=-1 to 0: the local maximum is at ( - 0.5, 5)? Wait, maybe the correct values are:

For interval \([-3, -1]\): the local minimum is at \(x=-2\), \(y = 0\).

For interval \([-1, 0]\): the local maximum is at the point (let's say x=-1, y=5? Wait, no, the graph shows a point at x=-1 to 0, the y - value is, from the graph, the local maximum in [-1,0] is 5? Wait, maybe the actual coordinates: when we hover, but since we can see the graph, the local minimum in [-3,-1] is 0 (at x=-2), local maximum in [-1,0] is 5 (at x=-1, y=5? Wait, no, the graph: at x=-2, y=0; then it goes up to a point (x=-1, y=5) then down to (0,0). Then from (0,0) down to (2, -30) then up.

For interval \([0, 3]\): the local minimum is at x=2, y=-30.

So:

1. Over \([-3, -1]\), local minimum is \(0\) (at \(x = - 2\)).

1. Over \([-1, 0]\), local maximum is \(5\)? Wait, no, maybe the point at x=-1 to 0 has y - coordinate 5? Wait, maybe the correct values are:

Wait, let's re - examine:

• Interval \([-3, -1]\): The graph has a minimum at \(x=-2\), \(y = 0\) (since it's the lowest in that interval).

• Interval \([-1, 0]\): The graph has a maximum at the point (let's say x=-1, y=5) but actually, looking at the graph, the point between -1 and 0: the local maximum is at \(x\) in [-1,0], and the \(y\) - value is, from the graph, the highest point there is at \(y = 5\) (maybe).

• Interval \([0, 3]\): The graph has a minimum at \(x = 2\), \(y=-30\) (the lowest point in that interval).

Answer:

Over the interval \([-3, -1]\), the local minimum is \(\boldsymbol{0}\).

Over the interval \([-1, 0]\), the local maximum is \(\boldsymbol{5}\) (assuming the point's y - coordinate is 5, based on the graph's grid).

Over the interval \([0, 3]\), the local minimum is \(\boldsymbol{-30}\).