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) = 4x2 − 8x + 3, [−1, 3] c =

Answer with explanation:Rolle's Theorem states that:If f is a continuous function in [a,b] and is differentiable in (a,b)such that f(a)=f(b)Then there exist a constant c in between a and b i.e. c∈[a,b]such that:  f'(c)=0Here we have the function f(x) as:[tex]f(x)=4x^2-8x+3[/tex] where x∈[-1,3]Since the function f(x) is a polynomial function hence it is continuous as well as differentiable over the interval [-1,3].Also,f(-1)=15(Since,[tex]f(-1)=4\times (-1)^2-8\times (-1)+3\\\\f(-1)=4+8+3\\\\f(-1)=15[/tex]   )and f(3)=15( Since,[tex]f(3)=4\times 3^2-8\times 3+3\\\\f(3)=36-24+3\\\\i.e.\\\\f(3)=15[/tex] )i.e. f(-1)=f(3)Hence, there will exist a c∈[-1,3] such that f'(c)=0[tex]f'(x)=8x-8\\\\i.e.\\\\f'(x)=0\\\\imply\\\\8x-8=0\\\\i.e.\\\\x-1=0\\\\i.e.\\\\x=1[/tex]Hence, the c that satisfy the conclusion is: c=1