Given f(x)=5-4 / x find all ralues of the in open interval (1,4) bich that f^prime(c)=(f(b)-f(a))/(b-a)

To find the values of in the open interval \( (1,4) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \), we first need to compute the derivative of \( f(x) \) and then find the points where the derivative equals the average rate of change over the interval.Given \( f(x) = 5 - \frac{4}{x} \), let's find \( f'(x) \): Now, we need to find the average rate of change of \( f(x) \) over the interval , where and : So, the average rate of change from to is: We need to find in the interval \( (1, 4) \) such that: Set the derivative equal to 1: Solve for : Therefore, the value of in the open interval \( (1, 4) \) that satisfies \( f'(c) = \frac{f(b) - f(a)}{b - a} \) is .