Question

which statement correctly identifies a local minimum of the graphed function?
over the interval -3, -2, the local minimum is 0.
over the interval -2, -1, the local minimum is 2.2.
over the interval -1, 0.5, the local minimum is 1.
over the interval 0.5, 2, the local minimum is 4.

Explanation:

Step1: Analyze Interval [-3, -2]

Check the function's values in [-3, -2]. The point (-2.2, 0) is on the interval? Wait, [-3, -2] and the graph's point (-2.2, 0) is near x=-2.2, but in [-3, -2], the function is decreasing? Wait, the graph at x=-2.2 (close to -2) is 0, but in [-3, -2], the minimum? Wait, the first option: Over [-3, -2], local min is 0. But the graph at x=-2.2 (which is in [-3, -2]?) Wait, x=-2.2 is less than -2, so in [-3, -2], the point (-2.2, 0) is there. But is that a local min? Wait, maybe not. Let's check other options.

Step2: Analyze Interval [-2, -1]

The interval [-2, -1]. The points here: at x=-2, y=0; then the graph goes up to (-1.3, 2.2). So the minimum in [-2, -1] would be 0, but the option says 2.2, which is a local max, so wrong.

Step3: Analyze Interval [-1, 0.5]

The interval [-1, 0.5]. The graph has a peak at (-1.3, 2.2) then decreases to (0, 1). So in [-1, 0.5], the minimum is at (0, 1), which is y=1. So this option says local min is 1, which matches.

Step4: Analyze Interval [0.5, 2]

The interval [0.5, 2]. The graph goes from (0, 1) up to (1, 4). So the minimum here is 1 (at x=0) but wait, x=0 is not in [0.5, 2]. Wait, in [0.5, 2], the graph starts at x=0.5 (which is near 0, y=1) and goes up to (1, 4). So the minimum in [0.5, 2] is 1? But the option says 4, which is a local max, so wrong.

Wait, let's recheck:

First option: [-3, -2]. The function at x=-3? Wait, the graph starts below x=-2, going up to x=-2 (y=0). So in [-3, -2], the minimum is the lowest point, which is at the end (x=-3, but we don't see that, but the point (-2.2, 0) is at x=-2.2 (in [-3, -2]? -2.2 is between -3 and -2? Yes, -3 < -2.2 < -2. So the minimum there is 0? But wait, maybe the first option is wrong. Wait, no, let's check the third option: [-1, 0.5]. The graph from x=-1 to x=0.5: at x=-1, y is around 2.2 (the peak), then decreases to (0, 1), then maybe increases? Wait, (0, 1) is the lowest in [-1, 0.5], so y=1. So the third option says local min is 1, which is correct.

Wait, the first option: Over [-3, -2], local min is 0. But in [-3, -2], the function is at x=-2.2 (y=0) and maybe lower? Wait, the graph is going up from the bottom (left) to x=-2 (y=0). So in [-3, -2], the minimum is 0? But maybe the first option is incorrect. Wait, let's check the third option again. The interval [-1, 0.5]. The graph has a point at (0, 1), which is in [-1, 0.5] (since 0 is between -1 and 0.5). So the minimum in that interval is 1, which is correct. The other options:

• Second option: [-2, -1], local min is 2.2. But 2.2 is a local max (the peak), so wrong.

• Fourth option: [0.5, 2], local min is 4. 4 is a local max, wrong.

• First option: [-3, -2], local min is 0. But in [-3, -2], the function is decreasing? Wait, from x=-3 to x=-2, the graph goes from lower (maybe -infinity) up to x=-2 (y=0). So the minimum in [-3, -2] would be the leftmost point, but we don't know, but the point (-2.2, 0) is at x=-2.2 (in [-3, -2]? -2.2 is greater than -3 and less than -2, so yes. But is that a local min? A local minimum is a point where the function is lower than its neighbors. At x=-2.2, y=0, and to the left (x < -2.2), the function is lower (since it's going up to x=-2.2), so x=-2.2 is a local min? But the first option says "over the interval [-3, -2], the local minimum is 0". But maybe the correct option is the third one: Over [-1, 0.5], local min is 1. Because in that interval, the function decreases from (-1.3, 2.2) to (0, 1), so the minimum is 1.

Yes, so the correct option is: Over the interval [-1, 0.5], the local minimum is 1.

Answer:

Over the interval [-1, 0.5], the local minimum is 1. (The option corresponding to this statement, which is the third option in the list: "Over the interval [-1, 0.5], the local minimum is 1.")