Please solve this---Question Number: 21 Chart/Diagram Description: Type: Graph of a function on an xy-plane. Coordinate Axes: X-axis labeled 'x', Y-axis labeled 'y'. The origin is labeled 'O'. X-axis Scale: Marked from -4 to 6, with labels at -4, -2, O (0), 2, 4, 6. Increments are shown every unit. Y-axis Scale: Marked from -2 to 4, with labels at -2, O (0), 2, 4. Increments are shown every unit. Curve: A continuous curve is plotted. It appears to start at approximately x = -4, goes upward, crosses the x-axis at approximately x = -2 and x = 0, reaches a peak between x=0 and x=1 (approximately at y=4), and then goes downward, ending at approximately x=6 with a y-value around 2. The question specifies the domain of interest as -4 <= x <= 6. Within this domain, the graph starts at x=-4 with a y-value that appears to be approximately -4. It then increases to a maximum and decreases. The minimum value on the interval [-4, 6] would occur at one of the endpoints or a local minimum within the interval. From the graph, the value at x=-4 appears to be the lowest point on the graph within the specified interval. Question Stem: What is the minimum value of the function graphed on the xy-plane above, for -4 <= x <= 6 ? Options: A) -infinity B) -4 C) -2 D) 1

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Let's find the minimum value of the function graphed on the xy-plane for x between negative 4 and 6. First, we identify the domain from negative 4 to 6, shown by the vertical dashed lines. Next, we examine the function values at the endpoints and any critical points. At x equals negative 4, the function value is negative 4. At x equals 6, the function value is approximately 2. Looking at the graph, we can see there are no local minima within the open interval. The function has a local maximum but no local minimum. Therefore, the minimum value must occur at one of the endpoints. Comparing the values, we find that the minimum value is negative 4, which occurs at x equals negative 4. The answer is B. Now let's analyze the critical points and endpoints of our function. At the left endpoint where x equals negative 4, the function value is negative 4. This is highlighted in red. At approximately x equals 1, we have a local maximum where the function value is about 4, shown in blue. At the right endpoint where x equals 6, the function value is approximately 2, marked in green. Since there are no local minima within the interval, the minimum must occur at one of the endpoints. Comparing the values at x equals negative 4 and x equals 6, we can see that negative 4 is smaller than 2. Therefore, the minimum value of the function on the interval from negative 4 to 6 is negative 4, which occurs at the left endpoint. This confirms our answer is B. Let's compare the function values at different points within our domain. We've calculated the function values at several key points: At x equals negative 4, the function value is negative 4. At x equals negative 2, the function value is approximately negative 0.5. At x equals 0, the function value is about 2.5. At x equals 1, which is near the local maximum, the function value is approximately 4. At x equals 3, the function value is about 3.2. And at x equals 6, the function value is approximately 2. Looking at all these values, we can clearly see that the minimum value is negative 4, which occurs at x equals negative 4. The horizontal dashed line shows this minimum value across the domain. Therefore, our answer is B, negative 4. Now let's analyze each of the answer choices. Option A suggests the minimum value is negative infinity. This is incorrect because our function has a finite minimum value within the given domain. The graph clearly shows the function doesn't extend below negative 4. Option B suggests the minimum value is negative 4. This is correct, as we've verified that the function reaches its minimum value of negative 4 at x equals negative 4, shown by the red horizontal line. Option C suggests the minimum value is negative 2. This is incorrect because we've already found that the function goes down to negative 4, which is less than negative 2. The gray line shows where y equals negative 2 would be. Option D suggests the minimum value is 1. This is also incorrect because the function goes well below 1 in our domain. Therefore, the correct answer is B: negative 4. Let's summarize what we've learned about finding the minimum value of a function on a closed interval. The minimum value of a function on a closed interval can occur either at the endpoints or at critical points within the interval. For the given function on the interval from negative 4 to 6, we identified that the minimum occurs at the left endpoint, x equals negative 4. The function value at x equals negative 4 is negative 4, which is lower than all other values in the interval. We verified that there are no local minima within the interval that could be lower than this value. Therefore, the minimum value of the function is negative 4, and the correct answer is option B. This problem demonstrates the importance of checking both endpoints and critical points when finding extreme values of a function on a closed interval.