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 thatf'(c)= f(b) - f(a) / b - a

Answer 1

Given:

The function

$f(x)=x^3-3x^2+9x+5,\text{ on interval}[0,1]$

Required:

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.

Explanation:

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

First evaluate the function at the endpoints of the interval:

$\begin{gathered} f(1)=12 \\ f(0)=5 \end{gathered}$

Next, find the derivative:

$f^{\prime}(c)=3c^2-6c+9$

From the equation:

$\begin{gathered} 3c^2-6c+9=\frac{12-5}{1-0} \\ Simplify: \\ 3c^2-6c+9=7 \\ Solve\text{ the equation on the given interval}: \\ c=1-\frac{\sqrt{3}}{3} \end{gathered}$

Answer:

Completed the answer.