Let r(x) = 2x - 8 continuous over the interval [4, 6]. Find r(4) and r(6). According to the Intermediate Value Theorem, can you find a value where r(c) = 0?

To find r(4) and r(6), we simply substitute the values of x into the function r(x): r(4) = 2(4) - 8 = 8 - 8 = 0 r(6) = 2(6) - 8 = 12 - 8 = 4 Now, let's consider the Intermediate Value Theorem. This theorem states that if a function f(x) is continuous on a closed interval [a, b] and if f(a) and f(b) have opposite signs, then there exists at least one c in (a, b) such that f(c) = 0. In this case, r(x) is continuous on the interval [4, 6], and r(4) = 0 and r(6) = 4. Since r(4) and r(6) have opposite signs, we can conclude that there exists at least one c in (4, 6) such that r(c) = 0. To find this value, we can use the Intermediate Value Theorem's corollary, which states that if f(a) and f(b) have opposite signs, then there exists a c in (a, b) such that f(c) = (f(a) + f(b)) / 2. Applying this corollary, we get: c = (4 + 6) / 2 = 10 / 2 = 5 Therefore, there exists a value c = 5 in the interval (4, 6) such that r(c) = 0.