To find the absolute maximum and minimum values of the function [tex]\( f(x) = 4 - 2x^2 \)[/tex] over the interval [tex]\([-5, 1]\)[/tex], follow these steps:
1. Evaluate the function at the endpoints of the interval:
- [tex]\( f(-5) = 4 - 2(-5)^2 = 4 - 2 \times 25 = 4 - 50 = -46 \)[/tex]
- [tex]\( f(1) = 4 - 2(1)^2 = 4 - 2 \times 1 = 4 - 2 = 2 \)[/tex]
2. Find the critical points by setting the derivative [tex]\( f'(x) \)[/tex] to zero and solving for [tex]\( x \)[/tex]:
- The derivative of [tex]\( f(x) = 4 - 2x^2 \)[/tex] is [tex]\( f'(x) = -4x \)[/tex]
- Setting the derivative equal to zero: [tex]\( -4x = 0 \)[/tex] implies [tex]\( x = 0 \)[/tex]
3. Evaluate the function at the critical point within the interval:
- [tex]\( f(0) = 4 - 2(0)^2 = 4 - 2 \times 0 = 4 \)[/tex]
4. Compare the function values at the endpoints and critical points:
- [tex]\( f(-5) = -46 \)[/tex]
- [tex]\( f(1) = 2 \)[/tex]
- [tex]\( f(0) = 4 \)[/tex]
5. Determine the absolute maximum and minimum values:
- The maximum value is [tex]\( 4 \)[/tex] and it occurs at [tex]\( x = 0 \)[/tex]
- The minimum value is [tex]\( -46 \)[/tex] and it occurs at [tex]\( x = -5 \)[/tex]
Thus, the absolute maximum value of [tex]\( f(x) \)[/tex] over the interval [tex]\([-5, 1]\)[/tex] is [tex]\( 4 \)[/tex] and this occurs at [tex]\( x = 0 \)[/tex]. The absolute minimum value is [tex]\( -46 \)[/tex] and this occurs at [tex]\( x = -5 \)[/tex].
