Answer: The Intermediate Value Theorem states that if a function is continuous on a closed interval and takes on two different values at the endpoints of the interval, then it must take on all values between those two endpoints.
Given the function f(x) = x^2 - x + 11 and the interval (2, 11), we know that f(2) = 2^2 - 2 + 11 = 13 and f(11) = 11^2 - 11 + 11 = 121. And we want to find the value of c such that f(c) = 31.
Since 31 is between 13 and 121 and the function f(x) is continuous on the interval (2, 11), we can conclude that by the Intermediate Value Theorem, there exists a value c in the interval (2, 11) such that f(c) = 31. To find the specific value of c, we would have to perform further calculations or use a root-finding method.
Explanation: