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.

Step-by-step 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)=(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=(12-5)/(1-0) \\ Simplify: \\ 3c^2-6c+9=7 \\ Solve\text{ the equation on the given interval}: \\ c=1-(√(3))/(3) \end{gathered}`

Answer:

Completed the answer.