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 \( c \) 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.<br /><br />Given \( f(x) = 5 - \frac{4}{x} \), let's find \( f'(x) \):<br /><br />\[ f'(x) = \frac{d}{dx} \left( 5 - \frac{4}{x} \right) = \frac{4}{x^2} \]<br /><br />Now, we need to find the average rate of change of \( f(x) \) over the interval \([a, b]\), where \( a = 1 \) and \( b = 4 \):<br /><br />\[ f(1) = 5 - \frac{4}{1} = 1 \]<br />\[ f(4) = 5 - \frac{4}{4} = 4 \]<br /><br />So, the average rate of change from \( x = 1 \) to \( x = 4 \) is:<br /><br />\[ \frac{f(4) - f(1)}{4 - 1} = \frac{4 - 1}{3} = 1 \]<br /><br />We need to find \( c \) in the interval \( (1, 4) \) such that:<br /><br />\[ f'(c) = 1 \]<br /><br />Set the derivative equal to 1:<br /><br />\[ \frac{4}{c^2} = 1 \]<br /><br />Solve for \( c \):<br /><br />\[ 4 = c^2 \]<br />\[ c^2 = 4 \]<br />\[ c = \sqrt{4} \]<br />\[ c = 2 \]<br /><br />Therefore, the value of \( c \) in the open interval \( (1, 4) \) that satisfies \( f'(c) = \frac{f(b) - f(a)}{b - a} \) is \( c = 2 \).