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 βˆ’ 12x + 3x2, [1, 3]

Accepted Solution

Answer:c=2Step-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)^2f(1)=1-12+3f(1)=-8f(3)=1-12(3)+3(3)^2f(3)=1-36+27f(3)=-8So 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^2f'(x)=0-12+6xf'(x)=-12+6xRemember we need yo solve f'(c)=0 for c.So we need to solve -12+6c=0.-12+6c=0Add 12 on both sides:6c=12Divide both sides by 6:c=2