verify that the function satisfies the three hypotheses of rolles theorem on the given interval. then find…

verify that the function satisfies the three hypotheses of rolles theorem on the given interval. then find all numbers c that satisfy the conclusion of rolles theorem. (enter your answer comma - separated list.)\n\n$f(x)=3x^{2}-6x + 4$, $-1,3$\n\n$c =$

verify that the function satisfies the three hypotheses of rolles theorem on the given interval. then find all numbers c that satisfy the conclusion of rolles theorem. (enter your answer comma - separated list.)\n\n$f(x)=3x^{2}-6x + 4$, $-1,3$\n\n$c =$

Answer

Explanation:

Step1: Check continuity

A polynomial function (y = 3x^{2}-6x + 4) is continuous everywhere, so it is continuous on the closed interval ([-1,3]).

Step2: Check differentiability

The derivative (f^\prime(x)=\frac{d}{dx}(3x^{2}-6x + 4)=6x-6) exists for all (x), so the function is differentiable on the open interval ((-1,3)).

Step3: Check (f(-1)=f(3))

Calculate (f(-1)=3\times(-1)^{2}-6\times(-1)+4=3 + 6+4 = 13) and (f(3)=3\times3^{2}-6\times3 + 4=27-18 + 4=13).

Step4: Find (c)

Set (f^\prime(c)=0). Since (f^\prime(x)=6x - 6), then (6c-6 = 0). Solve for (c): [ \begin{align*} 6c-6&=0\ 6c&=6\ c&=1 \end{align*} ]

Answer:

(1)