First, let's consider the function f(x) = -x^2 + 7x which is continuous on the closed interval [4,9]. We want to apply the Intermediate Value Theorem (IVT), a fundamental theorem in calculus, to this function. The IVT states that if a function is continuous on a closed interval [a,b] and K is any number between f(a) and f(b), then there exists a number c in [a,b] such that f(c) = K.
For our example, we need to first calculate the value of our function at the endpoints of the interval, f(4) and f(9). By substituting these values into our function, we find that f(4) = 12.0 and f(9) = -18.0.
Next, we want to verify whether the value 10 lies between the values of f(4) and f(9). Since 10 does fall between f(4) = 12.0 and f(9) = -18.0, we can apply the IVT which confirms the existence of a value c in the interval [4,9] such that f(c) = 10.
To find the exact value of c we then solve the equation f(x) = 10, which is equivalent to -x^2 + 7x = 10. After solving this equation, we find that x = 5 is a solution.
However, we need to ensure that this solution falls within our given interval [4,9]. So, we verify that 5 lies within this interval. Therefore, we can now confirm not only the existence of a solution, based on the IVT, but also the specific value, c = 5.