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 answers as a comma - separated list.)\n( f(x)=3x^{2}-6x + 4,-1,3 )

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 answers as a comma - separated list.)\n( f(x)=3x^{2}-6x + 4,-1,3 )

Answer

Explanation:

Step1: Check continuity

Since (f(x)=3x^{2}-6x + 4) is a polynomial, it is continuous on ([-1,3]).

Step2: Check differentiability

The derivative (f^\prime(x)=6x - 6) exists for all (x), so (f(x)) is differentiable on ((-1,3)).

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

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

Step4: Find (c)

Set (f^\prime(c)=0), i.e., (6c-6 = 0) (6c=6) (c = 1)

Answer:

(1)