solve this---21 Chart Description: * Type: Line chart showing the graph of a function. * Coordinate Axes: X-axis labeled 'x', Y-axis labeled 'y'. The origin is labeled 'O'. * Scales: Both axes have tick marks. The X-axis ticks are labeled -4, -2, O, 2, 4, 6. The Y-axis ticks are labeled -2, O, 2, 4. Each major tick mark represents a unit of 2. There are minor tick marks every unit of 1. * The Curve: A curved line starting at approximately x = -4, increasing steeply, reaching a peak around x = 0.5 and y = 4, and then decreasing gradually towards x = 6. The curve is shown within the range approximately from x = -4 to x = 6. Question Stem: What is the minimum value of the function graphed on the xy-plane above, for -4 ≤ x ≤ 6 ? Options: A) -∞ B) -4 C) -2 D) 1

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In this problem, we need to find the minimum value of the function shown in the graph, where x is between negative 4 and positive 6. Let's analyze the graph to identify the lowest point within this domain. Looking at the graph, we can see that the function starts at around negative 2 when x equals negative 4, rises to a peak of about 4 near x equals 0.5, and then decreases to negative 2 at x equals 6. The minimum value appears to occur at x equals 6, where the y-value is negative 2. Let's analyze the function's behavior across the given domain from negative 4 to positive 6. At x equals negative 4, the function value is approximately negative 1.9. As x increases, the function rises to a maximum of about 4 at x equals 0.5, and then decreases. At x equals 6, the function reaches its lowest value within our domain, which is negative 2. By comparing all values within the domain, we can confirm that negative 2 is indeed the minimum value of the function for x between negative 4 and 6. Now let's evaluate each answer option. Option A suggests the minimum value is negative infinity, but our graph clearly shows a finite minimum value. Option B suggests negative 4, but our analysis shows the minimum is negative 2, which is greater than negative 4. Option C suggests negative 2, which exactly matches the minimum value we found at x equals 6. Option D suggests 1, but our minimum value is less than 1. Therefore, the correct answer is C, negative 2. Let's summarize our approach to finding the minimum value of this function. We analyzed the function over the given domain from x equals negative 4 to x equals 6. We identified key points on the graph: at x equals negative 4, the y-value is approximately negative 1.9; at x equals 0.5, the function reaches its maximum of about 4; and at x equals 6, the function reaches its minimum value of negative 2 within our domain. By comparing all values across the domain, we confirmed that negative 2 is indeed the minimum value. Therefore, the correct answer is option C, negative 2. To conclude, let's review the key strategies for finding minimum or maximum values from a graph. First, always identify the domain constraints, which in this case was x between negative 4 and 6. Second, examine all critical points within that domain, including where the derivative equals zero. Third, check the function values at the endpoints of the domain. Finally, compare all these values to determine the minimum. For this specific problem, we found that the minimum value occurs at x equals 6, where the function equals negative 2. Therefore, the correct answer is option C, negative 2.