using the intermediate value theorem, determine, if possible, whether the function f has at least one real…

using the intermediate value theorem, determine, if possible, whether the function f has at least one real zero between a and b.\n\nf(x)=x³ + 5x² - 5x - 1; a = -8, b = -5\n\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\n○ 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.)\n\n○ 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.)\n\n○ c. it is impossible to use the intermediate value theorem in this case.

using the intermediate value theorem, determine, if possible, whether the function f has at least one real zero between a and b.\n\nf(x)=x³ + 5x² - 5x - 1; a = -8, b = -5\n\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\n○ 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.)\n\n○ 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.)\n\n○ c. it is impossible to use the intermediate value theorem in this case.

Answer

Explanation:

Step1: Calculate ( f(a) )

Substitute ( a = - 8 ) into ( f(x)=x^{3}+5x^{2}-5x - 1 ). [ \begin{align*} f(-8)&=(-8)^{3}+5\times(-8)^{2}-5\times(-8)-1\ &=-512 + 5\times64+40 - 1\ &=-512+320 + 40-1\ &=-153 \end{align*} ]

Step2: Calculate ( f(b) )

Substitute ( b=-5 ) into ( f(x)=x^{3}+5x^{2}-5x - 1 ). [ \begin{align*} f(-5)&=(-5)^{3}+5\times(-5)^{2}-5\times(-5)-1\ &=-125+5\times25 + 25-1\ &=-125 + 125+25-1\ &=24 \end{align*} ] 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 ).