To determine the absolute maximum of the function [tex]\( f(x) = x^2 - 12 \)[/tex] on the interval [tex]\([-3, 4]\)[/tex], follow these steps:
1. Identify the critical points:
- You need to find where the derivative of the function [tex]\(f\)[/tex] is zero or does not exist within the interval.
- The derivative of [tex]\( f(x) \)[/tex] is [tex]\( f'(x) = 2x \)[/tex].
- Solve [tex]\( 2x = 0 \)[/tex] to find the critical points:
[tex]\[
2x = 0 \implies x = 0.
\][/tex]
- Check if the critical point lies within the interval [tex]\([-3, 4]\)[/tex]. In this case, [tex]\( x = 0 \)[/tex] is within the interval.
2. Evaluate the function at the endpoints and the critical points:
- Evaluate [tex]\(f(x)\)[/tex] at the critical point [tex]\(x = 0\)[/tex]:
[tex]\[
f(0) = 0^2 - 12 = -12.
\][/tex]
- Evaluate [tex]\(f(x)\)[/tex] at the endpoints [tex]\(x = -3\)[/tex] and [tex]\(x = 4\)[/tex]:
[tex]\[
f(-3) = (-3)^2 - 12 = 9 - 12 = -3,
\][/tex]
[tex]\[
f(4) = 4^2 - 12 = 16 - 12 = 4.
\][/tex]
3. Compare the values from the endpoints and critical points:
- At [tex]\( x = 0 \)[/tex], [tex]\( f(0) = -12 \)[/tex].
- At [tex]\( x = -3 \)[/tex], [tex]\( f(-3) = -3 \)[/tex].
- At [tex]\( x = 4 \)[/tex], [tex]\( f(4) = 4 \)[/tex].
The highest value among these evaluations determines the absolute maximum.
4. Conclusion:
- The absolute maximum value of [tex]\( f \)[/tex] on the interval [tex]\([-3, 4]\)[/tex] is [tex]\( 4 \)[/tex] at [tex]\( x = 4 \)[/tex].
Therefore, the correct choice is:
A. The absolute maximum is [tex]\(4\)[/tex] at [tex]\( x = 4 \)[/tex].
