Question

determine whether the Mean Value Theorem can be applied to f on the closed interval [a,b]. If the Mean Value Theorem can be applied, find all values of in the open interval such that

Answer 1

Given:

The function

$f(x)=\frac{x}{x-5},\text{ on interval}[1,4]$

Required:

Dermine whether the Mean Value Theorem can be applied to f on the closed interval [a,b]. If the Mean Value Theorem can be applied, find all values of in the open interval .

Explanation:

$\begin{gathered} \text{The mean value theorem states that for a continuous and differentiable} \\ \text{ function }f(x)\text{ on the interval }[a,b],\text{ there exists such number }c\text{ from } \\ \text{ interval }(a,b),\text{ that }f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}. \end{gathered}$

First, evaluate the function at the endpoints of the interval:

$\begin{gathered} f(4)=-4 \\ f(1)=-\frac{1}{4} \end{gathered}$

Next, find the derivative

$\begin{gathered} f^{\prime}(c)=-\frac{c}{(c-5)^2}+\frac{1}{c-5} \\ =\frac{(-4)-(-\frac{1}{4})}{(4)-(1)} \\ Simplify: \\ -\frac{c}{(c-5)^2}+\frac{1}{c-5}=-\frac{5}{4} \\ Sol\text{ve the equation on the given interval}:c=3 \end{gathered}$

Answer:

The value of c = 3.