determine whether rolles theorem applies to the following function on the given interval. if so, find the…

determine whether rolles theorem applies to the following function on the given interval. if so, find the point(s) that are guaranteed to exist by rolles theorem. g(x)=x^3 - 3x^2 - 9x - 5; -1,5 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. rolles theorem applies and the point(s) guaranteed to exist is/are x = (type an exact answer, using radicals as needed. use a comma to separate answers as needed.) b. rolles theorem does not apply.

determine whether rolles theorem applies to the following function on the given interval. if so, find the point(s) that are guaranteed to exist by rolles theorem. g(x)=x^3 - 3x^2 - 9x - 5; -1,5 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. rolles theorem applies and the point(s) guaranteed to exist is/are x = (type an exact answer, using radicals as needed. use a comma to separate answers as needed.) b. rolles theorem does not apply.

Answer

Explanation:

Step1: Check continuity

The function $g(x)=x^{3}-3x^{2}-9x - 5$ is a polynomial function. Polynomial functions are continuous everywhere, so $g(x)$ is continuous on the closed - interval $[-1,5]$.

Step2: Check differentiability

The derivative of $g(x)$ using the power rule $(x^n)'=nx^{n - 1}$ is $g'(x)=3x^{2}-6x - 9$. Since $g'(x)$ exists for all real $x$, $g(x)$ is differentiable on the open interval $(-1,5)$.

Step3: Evaluate function at endpoints

$g(-1)=(-1)^{3}-3(-1)^{2}-9(-1)-5=-1 - 3 + 9 - 5=0$. $g(5)=5^{3}-3\times5^{2}-9\times5 - 5=125-75 - 45 - 5=0$. Since $g(-1)=g(5)=0$, Rolle's Theorem applies.

Step4: Find critical points

Set $g'(x)=0$. So $3x^{2}-6x - 9 = 0$. Divide through by 3: $x^{2}-2x - 3=0$. Factor the quadratic equation: $(x - 3)(x+1)=0$. Solve for $x$: $x = 3$ or $x=-1$. But we are looking for values in the open interval $(-1,5)$, so we discard $x=-1$.

Answer:

A. Rolle's Theorem applies and the point(s) guaranteed to exist is/are $x = 3$