Answer :
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].
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].
