To determine if the function [tex]\( f(x) = 3x^2 - 5x - 1 \)[/tex] has at least one real zero between [tex]\( a = -2 \)[/tex] and [tex]\( b = -1 \)[/tex] using the Intermediate Value Theorem (IVT), follow these steps:
1. Determine [tex]\( f(a) \)[/tex]:
- Substitute [tex]\( a = -2 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-2) = 3(-2)^2 - 5(-2) - 1 \][/tex]
[tex]\[ f(-2) = 3(4) + 10 - 1 \][/tex]
[tex]\[ f(-2) = 12 + 10 - 1 = 21 \][/tex]
2. Determine [tex]\( f(b) \)[/tex]:
- Substitute [tex]\( b = -1 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(-1) = 3(-1)^2 - 5(-1) - 1 \][/tex]
[tex]\[ f(-1) = 3(1) + 5 - 1 \][/tex]
[tex]\[ f(-1) = 3 + 5 - 1 = 7 \][/tex]
3. Apply the Intermediate Value Theorem:
- The IVT states that if a function [tex]\( f \)[/tex] is continuous on the interval [tex]\([a, b]\)[/tex] and [tex]\( f(a) \)[/tex] and [tex]\( f(b) \)[/tex] have opposite signs, there is at least one real zero between [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. In this case, [tex]\( f(-2) = 21 \)[/tex] and [tex]\( f(-1) = 7 \)[/tex], both of which are positive.
Since [tex]\( f(-2) = 21 \)[/tex] and [tex]\( f(-1) = 7 \)[/tex] do not have opposite signs (both are positive), according to the IVT, the function [tex]\( f \)[/tex] does not have at least one real zero between [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
Therefore, the correct choice is:
A. By the intermediate value theorem, the function does not have at least one real zero between [tex]\( a \)[/tex] and [tex]\( b \)[/tex] because [tex]\( f(a) = 21 \)[/tex] and [tex]\( f(b) = 7 \)[/tex].
