Question

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.)

$f(x)=3x^{2}-6x + 4$, $-1,3$

$c =$

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 \(c\):

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Answer:

\(1\)