watch the video and then solve the problem given below.\nclick here to watch the video.\nusing the…

watch the video and then solve the problem given below.\nclick here to watch the video.\nusing the intermediate value theorem, determine, if possible, whether the function f has at least one real zero between a and b.\nf(x)=x³ + 2x² - 7x - 1; a = -5, b = -3\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\na. the function does not have at least one real zero between a and b because f(a)= and f(b)=. (simplify your answers.)\nb. the function has at least one real zero between a and b because f(a)= and f(b)=. (simplify your answers.)\nc. it is impossible to use the intermediate value theorem in this case.

watch the video and then solve the problem given below.\nclick here to watch the video.\nusing the intermediate value theorem, determine, if possible, whether the function f has at least one real zero between a and b.\nf(x)=x³ + 2x² - 7x - 1; a = -5, b = -3\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\na. the function does not have at least one real zero between a and b because f(a)= and f(b)=. (simplify your answers.)\nb. the function has at least one real zero between a and b because f(a)= and f(b)=. (simplify your answers.)\nc. it is impossible to use the intermediate value theorem in this case.

Answer

Explanation:

Step1: Calculate (f(a))

Substitute (a = - 5) into (f(x)=x^{3}+2x^{2}-7x - 1). [ \begin{align*} f(-5)&=(-5)^{3}+2\times(-5)^{2}-7\times(-5)-1\ &=-125 + 2\times25+35 - 1\ &=-125+50 + 35-1\ &=-41 \end{align*} ]

Step2: Calculate (f(b))

Substitute (b=-3) into (f(x)=x^{3}+2x^{2}-7x - 1). [ \begin{align*} f(-3)&=(-3)^{3}+2\times(-3)^{2}-7\times(-3)-1\ &=-27+2\times9 + 21-1\ &=-27 + 18+21-1\ &=11 \end{align*} ]

Step3: Apply the Intermediate - Value Theorem

Since (f(x)) is a polynomial, it is continuous on the interval ([-5,-3]). And (f(-5)=-41) and (f(-3)=11), so (f(-5)\cdot f(-3)=(-41)\times11=-451<0).

Answer:

B. The function has at least one real zero between (a) and (b) because (f(a)=-41) and (f(b)=11)