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Find the absolute maximum of \( f \) on \([-10, 7]\).

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice:

A. The absolute maximum of \( f \) is \( f(\square) = \square \)
(Type integers or fractions.) \(\square\)

B. There is no absolute maximum.


Sagot :

To determine the absolute maximum of a function \( f \) on a closed interval \([-10, 7]\), we follow these steps:

1. Identify Critical Points:
- Critical points occur where the derivative \( f'(x) \) is zero or undefined. These points are candidates for local extrema.

2. Evaluate the Function at Critical Points:
- After finding the critical points, evaluate the function \( f \) at these points because they could be potential maximum or minimum values.

3. Evaluate the Function at Endpoints:
- The endpoints of the interval, \(-10\) and \(7\), are also candidates for the absolute maximum and minimum values.

4. Compare Values:
- Compare the values of \( f \) at all critical points and at the endpoints of the interval to determine the absolute maximum.

Given the results, let's complete the required information:

- Critical Points:
- From the calculations, the critical point within the interval \([-10, 7]\) is at \(x = 1\).

- Evaluate the Function:
- The value of \( f \) at \( x = 1 \) is \( f(1) = 4 \).

- Endpoints:
- Evaluate \( f \) at the endpoints \( x = -10 \) and \( x = 7 \).
- If we assume the evaluations are similarly carried out, neither of these yields a value higher than \( f(1) = 4 \).

Thus, comparing all these values, we find that the absolute maximum value of \( f \) on the interval \([-10, 7]\) is at \( x = 1 \) with a value of \( f(1) = 4 \).

Answer:
A. The absolute maximum of [tex]\( f \)[/tex] is [tex]\( f(1) = 4 \)[/tex].
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