To determine if [tex]\(-1\)[/tex] is a lower bound for the zeros of the function [tex]\( f(x) = x^4 + x^3 - 11x^2 - 9x + 18 \)[/tex], we need to evaluate the function at [tex]\( x = -1 \)[/tex].
1. Substitute [tex]\(-1\)[/tex] into the function:
[tex]\( f(-1) = (-1)^4 + (-1)^3 - 11(-1)^2 - 9(-1) + 18 \)[/tex]
2. Calculate each term:
- [tex]\( (-1)^4 = 1 \)[/tex]
- [tex]\( (-1)^3 = -1 \)[/tex]
- [tex]\( -11(-1)^2 = -11(1) = -11 \)[/tex]
- [tex]\( -9(-1) = 9 \)[/tex]
- [tex]\( 18 \)[/tex]
3. Add these values together:
[tex]\( f(-1) = 1 + (-1) - 11 + 9 + 18 \)[/tex]
4. Simplify the expression:
[tex]\( f(-1) = 1 - 1 - 11 + 9 + 18 \)[/tex]
Combining these step-by-step:
- [tex]\( 1 - 1 = 0 \)[/tex]
- [tex]\( 0 - 11 = -11 \)[/tex]
- [tex]\( -11 + 9 = -2 \)[/tex]
- [tex]\( -2 + 18 = 16 \)[/tex]
5. Evaluate the function:
[tex]\( f(-1) = 16 \)[/tex]
The value of [tex]\( f(-1) \)[/tex] is [tex]\( 16 \)[/tex], which is greater than zero.
Since [tex]\( f(-1) > 0 \)[/tex], this indicates that [tex]\( -1 \)[/tex] is not a lower bound for the zeros of the function because a lower bound would require [tex]\( f(x) \)[/tex] to be non-positive (i.e., [tex]\( \leq 0 \)[/tex]) at that point.
Thus, the correct answer is B. False.
