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Answer :
To determine the minimum cost on the interval for the given cost function [tex]\( C(q) = 0.1q^3 - 14.85q^2 + 688.2q + 300 \)[/tex], we need to follow these steps:
1. Identify the Interval: The problem specifies that we want to find the minimum cost on the interval where [tex]\( q \)[/tex] is between certain boundaries. In this case, you should check interval endpoints.
2. Find the Endpoints of the Interval: In optimization problems, checking the endpoints of the interval is essential. According to regular procedures, we should consider these from the options given. The values at the endpoints are 30 and 70.
3. Find the Critical Points: Critical points happen where the derivative of the cost function equals zero or is undefined. These are points where the slope of the tangent to the curve is flat, and they potentially indicate local minima or maxima.
4. Analyzing the Given Choices: To find where the cost might be minimized, we’ll evaluate the cost at both the endpoints and the critical points. Among the given options, points relevant for consideration include:
- 30 (the left endpoint)
- Critical points within the interval (which have been identified as 37, 49.5, and 62)
- 70 (the right endpoint)
5. List all Important Points to Check:
- Include the interval endpoints: 30 and 70.
- Include all critical points within the interval: 37, 49.5, and 62.
6. Conclusion: From the analysis of the options, the important values you need to check to find where the cost reaches its minimum on this interval are 30, 37, 49.5, 62, and 70.
By evaluating the cost at these points, you'll be able to determine the point where the cost is minimized in the given range.
1. Identify the Interval: The problem specifies that we want to find the minimum cost on the interval where [tex]\( q \)[/tex] is between certain boundaries. In this case, you should check interval endpoints.
2. Find the Endpoints of the Interval: In optimization problems, checking the endpoints of the interval is essential. According to regular procedures, we should consider these from the options given. The values at the endpoints are 30 and 70.
3. Find the Critical Points: Critical points happen where the derivative of the cost function equals zero or is undefined. These are points where the slope of the tangent to the curve is flat, and they potentially indicate local minima or maxima.
4. Analyzing the Given Choices: To find where the cost might be minimized, we’ll evaluate the cost at both the endpoints and the critical points. Among the given options, points relevant for consideration include:
- 30 (the left endpoint)
- Critical points within the interval (which have been identified as 37, 49.5, and 62)
- 70 (the right endpoint)
5. List all Important Points to Check:
- Include the interval endpoints: 30 and 70.
- Include all critical points within the interval: 37, 49.5, and 62.
6. Conclusion: From the analysis of the options, the important values you need to check to find where the cost reaches its minimum on this interval are 30, 37, 49.5, 62, and 70.
By evaluating the cost at these points, you'll be able to determine the point where the cost is minimized in the given range.
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