Question

watch the video and then solve the problem given below.
click here to watch the video.
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 + 2x² - 7x - 1; a = -5, b = -3
select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the function does not have at least one real zero between a and b because f(a)= and f(b)=. (simplify your answers.)
b. 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 \(a = - 5\) into \(f(x)=x^{3}+2x^{2}-7x - 1\).

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Step2: Calculate \(f(b)\)

Substitute \(b=-3\) into \(f(x)=x^{3}+2x^{2}-7x - 1\).

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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 \(f(b)=11\)