Using the intermediate value theorem, determine if the function [tex]f[/tex] has at least one real zero between [tex]a[/tex] and [tex]b[/tex].

[tex]\[ f(x) = x^3 + 5x^2 - 9x - 37 \][/tex]
[tex]\[ a = -9, \, b = -3 \][/tex]

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

A. By the intermediate value theorem, the function has at least one real zero between [tex]a[/tex] and [tex]b[/tex] because [tex]f(a) = \square[/tex] and [tex]f(b) = \square[/tex]. (Simplify your answers.)

B. 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) = \square[/tex] and [tex]f(b) = \square[/tex]. (Simplify your answers.)

C. It is impossible to use the intermediate value theorem in this case.



Answer :

To determine whether the function [tex]\( f(x) = x^3 + 5x^2 - 9x - 37 \)[/tex] has at least one real zero between [tex]\( a = -9 \)[/tex] and [tex]\( b = -3 \)[/tex] using the Intermediate Value Theorem, we proceed as follows:

1. Evaluate the Function at the Endpoints:
Calculate [tex]\( f(a) \)[/tex] and [tex]\( f(b) \)[/tex].

- For [tex]\( a = -9 \)[/tex]:
[tex]\[ f(-9) = (-9)^3 + 5(-9)^2 - 9(-9) - 37 \][/tex]
Simplifying step-by-step:
[tex]\[ (-9)^3 = -729 \][/tex]
[tex]\[ 5(-9)^2 = 5 \times 81 = 405 \][/tex]
[tex]\[ -9(-9) = 81 \][/tex]
Putting it all together:
[tex]\[ -729 + 405 + 81 - 37 = -280 \][/tex]
Thus, [tex]\( f(-9) = -280 \)[/tex].

- For [tex]\( b = -3 \)[/tex]:
[tex]\[ f(-3) = (-3)^3 + 5(-3)^2 - 9(-3) - 37 \][/tex]
Simplifying step-by-step:
[tex]\[ (-3)^3 = -27 \][/tex]
[tex]\[ 5(-3)^2 = 5 \times 9 = 45 \][/tex]
[tex]\[ -9(-3) = 27 \][/tex]
Putting it all together:
[tex]\[ -27 + 45 + 27 - 37 = 8 \][/tex]
Thus, [tex]\( f(-3) = 8 \)[/tex].

2. Apply the Intermediate Value Theorem (IVT):
The Intermediate Value Theorem states that if [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, then there exists at least one [tex]\( c \)[/tex] in the interval [tex]\((a, b)\)[/tex] such that [tex]\( f(c) = 0 \)[/tex].

In this case, [tex]\( f(-9) = -280 \)[/tex] and [tex]\( f(-3) = 8 \)[/tex]. Since [tex]\(-280\)[/tex] and [tex]\(8\)[/tex] have opposite signs, there is a change in sign from negative to positive. Therefore, by the Intermediate Value Theorem, there must be at least one root (zero) between [tex]\( a = -9 \)[/tex] and [tex]\( b = -3 \)[/tex].

Thus, the correct choice is:
[tex]\[ \text{A. By the intermediate value theorem, the function has at least one real zero between } a \text{ and } b \text{ because } f(a)= -280.00 \text{ and } f(b)= 8.00 \][/tex]