determine whether rolles theorem applies to the function (f(x)=-x^{2}+12x - 4) on the interval (2,10). if it…

determine whether rolles theorem applies to the function (f(x)=-x^{2}+12x - 4) on the interval (2,10). if it applies, find all possible values of (c) as in the conclusion of the theorem. separate multiple answers with a comma. if the theorem does not apply, write na for your answer.

determine whether rolles theorem applies to the function (f(x)=-x^{2}+12x - 4) on the interval (2,10). if it applies, find all possible values of (c) as in the conclusion of the theorem. separate multiple answers with a comma. if the theorem does not apply, write na for your answer.

Answer

Explanation:

Step1: Check continuity

The function $f(x)=-x^{2}+12x - 4$ is a polynomial. Polynomials are continuous everywhere, so $f(x)$ is continuous on the closed interval $[2,10]$.

Step2: Check differentiability

The derivative of $f(x)$ is $f^\prime(x)=-2x + 12$. Since $f^\prime(x)$ exists for all real - numbers, $f(x)$ is differentiable on the open interval $(2,10)$.

Step3: Evaluate $f(2)$ and $f(10)$

$f(2)=-(2)^{2}+12\times2 - 4=-4 + 24-4=16$. $f(10)=-(10)^{2}+12\times10 - 4=-100 + 120-4 = 16$. Since $f(2)=f(10)$, Rolle's Theorem applies.

Step4: Find $c$

Set $f^\prime(c)=0$. We have $f^\prime(c)=-2c + 12 = 0$. Solve for $c$: $-2c=-12$, so $c = 6$.

Answer:

$6$