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Given g(x)=−2x^2/x−1, find the absolute maximum value over the interval [−2,4].

a) -3
b) 6
c) 8
d) None of the above

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Final answer:

The maximum value of the function g(x) over the interval [-2, 4] is not provided directly in the given options. To find it, one must evaluate the function's endpoints and critical points. The proper conclusion is that there is no absolute maximum in the interval as the function increases indefinitely as x approaches 1, so the answer is (d) none of the above.

Step-by-step explanation:

To find the absolute maximum value of the function g(x) = -2x2 / (x - 1) over the interval [-2, 4], we need to perform a few steps. First, we must check the edges of the interval and the critical points where the derivative of the function is zero or undefined within the interval.

First, we find the value of the function at the interval endpoints:

  • g(-2) = -2(-2)2 / (-2 - 1) = -8 / -3 = 8/3
  • g(4) = -2(4)2 / (4 - 1) = -32 / 3

Then, we find the derivative of the function:

g'(x) = d/dx [-2x2 / (x - 1)]

Set the derivative equal to zero and solve for x, as well as check for where the derivative is undefined (when x = 1). After finding these critical points, evaluate the function at each critical point within the interval. However, we need to be careful because x = 1 is a point of discontinuity and cannot be a maximum.

Comparing the function's values at the critical points and at the interval's endpoints will allow us to identify the absolute maximum.

In case there is no clear maximum value from the critical points or endpoints within the given interval, we would conclude that the maximum value is either one of the given options (a) -3, (b) 6, (c) 8, or (d) none of the above.

Since the provided information about a, b, and c and the quadratic equation does not correlate directly with the function g(x), we cannot use it for this problem. Therefore, the correct answer is (d) none of the above since the function does not have an absolute maximum value on the given interval, as it increases indefinitely as we approach x = 1 from the left side.

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