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Answer :
Final answer:
The correct interpretation of f(-3) = 57 for the function f(x) = 5x^2 - 2x + 6 is 'None of the above'. This value simply informs us that at x = -3, the function's output is 57, and does not imply any of the specific characteristics of the function suggested by the options. The correct answer for the student's question is option (D).
Explanation:
The question asks to interpret the meaning of f(-3) = 57 for the function f(x) = 5x^2 - 2x + 6. The correct interpretation among the given choices is None of the above.
Here's why:
- Option A suggests that the value represents the average rate of change of f from x = -3 to x = 0. However, the average rate of change is calculated differently and requires evaluating the function at two points and finding the ratio of the change in f(x) to the change in x.
- Option B implying that 57 is the instantaneous rate of change at x = -3. This is incorrect as well since the instantaneous rate of change is given by the derivative of the function at that point, not the function's value itself.
- Option C says that it is the area under the curve from x = -3 to x = 0, which is also incorrect because the function value f(-3) does not represent the area under the curve; this would require integration.
Therefore, the correct interpretation is Option D: None of the above. The value of f(-3) = 57 simply tells us that when x is -3, the output of the function f(x) is 57.
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