find all numbers c that satisfy the conclusion of the Mean Value Theorem for f(x)= 3x^2 -4x+1

Answer 1

The mean value theorem requires the function to be continuous on a closed interval [a,b] and differentiable on the open interval with the same bounds. The good thing is that polynomials satisfy this quality for all real values of x.

Next step is to find the derivative of the function given and set f'(x) equal to [f(b)-f(a)]/(b-a)

The value of c are those which fall between a and b.

You need a specific interval to solve this problem.

Answer 2

Since no interval is given, we can only determine the general case in terms of a and b, b>a.

f'(c) = (f(b)-f(a))/(b-a)

f'(c) = 6c-4

(f(b)-f(a))/(b-a) = (3b²-4b+1 - 3a²+4a-1)/(b-a) = (3(b²-a²)-4(b-a))/(b-a) = 3(b+a)-4 = 6c-4

6c = 3(b+a)

c = (b+a)/2 The midpoint of an interval in this case.