Final answer:
The value of c guaranteed by the mean value theorem for the function f(x)=x^2 is c = (a + b)/2.
The correct option is c.
Step-by-step explanation:
The value of c guaranteed by the mean value theorem for the function f(x)=x^2 is c = (a + b)/2.
The mean value theorem states that for a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one value c in the open interval (a, b)
such that the derivative of the function at c is equal to the average rate of change of the function over the interval [a, b].
In this case, the function f(x)=x^2 is continuous and differentiable for any interval (a, b], so the value of c guaranteed by the mean value theorem is c = (a + b)/2.
The correct option is c.