Question

using the intermediate value theorem, determine, if possible, whether the function f has at least one real zero between a and b.

f(x)=x³ + 5x² - 5x - 1; a = -8, b = -5

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 does not have at least one real zero between a and b because f(a) = and f(b) = (simplify your answers.)

○ b. by the intermediate value theorem, the function has at least one real zero between a and b because f(a) = and f(b) = (simplify your answers.)

○ c. it is impossible to use the intermediate value theorem in this case.

Explanation:

Step1: Calculate \( f(a) \)

Substitute \( a = - 8 \) into \( f(x)=x^{3}+5x^{2}-5x - 1 \).

$$
LATEXBLOCK0
$$

Step2: Calculate \( f(b) \)

Substitute \( b=-5 \) into \( f(x)=x^{3}+5x^{2}-5x - 1 \).

$$
LATEXBLOCK1
$$

Since \( f(x) \) is a polynomial function, it is continuous everywhere. And \( f(-8)=-153<0 \), \( f(-5)=24>0 \).

Answer:

B. By the intermediate value theorem, the function has at least one real zero between \( a \) and \( b \) because \( f(a)=-153 \) and \( f(b)=24 \).