Verify that the function satisfies the three hypotheses of rolle's theorem on the given interval. then find all numbers c that satisfy the conclusion of rolle's theorem. (enter your answers as a comma-separated list.) f(x) = 1 − 12x + 3x2, [1, 3]

c=2

Step-by-step explanation:

All polynomials are continuous and differentiable on the set of real numbers.

So we just need to confirm on the given interval that f(1)=f(3).

f(1)=1-12(1)+3(1)^2

f(1)=1-12+3

f(1)=-8

f(3)=1-12(3)+3(3)^2

f(3)=1-36+27

f(3)=-8

So this means there is c in (1,3) such that f'(c)=0.

To solve that equation we must differentiate.

f(x)=1-12x+3x^2

f'(x)=0-12+6x

f'(x)=-12+6x

Remember we need yo solve f'(c)=0 for c.

So we need to solve -12+6c=0.

-12+6c=0

Add 12 on both sides:

6c=12

Divide both sides by 6:

c=2