Question

I’m not sure what this mean for #5 thanks for the help!

Answer 1

Answer:

c = 2.35

Step-by-step explanation:

The Mean Value Theorem is defined as:

`\begin{gathered} f^(\prime)(c)=(f(b)-f(a))/(b-a) \\ \\ \text{and} \\ f(c)=(1)/(b-a)\int ^4_1f(x)dx \end{gathered}`

With the given f(x), we have:

`\begin{gathered} f(c)=(1)/(b-a)\int ^4_1(x)/(x+2)dx \\ \\ f(c)=(1)/(4-1)\int ^4_1(1-(2)/(x+2))dx \\ \\ =(1)/(3)\lbrack x-2\ln (x+2)\rbrack\begin{cases}4 \\ 1\end{cases} \\ \\ =(1)/(3)\lbrack(4-2\ln (4+2))-(1-2\ln (1+2))\rbrack \\ \\ =1-(2)/(3)\ln (2) \\ \\ =0.54 \end{gathered}`

We have

`\begin{gathered} f(c)=0.54 \\ (c)/(c+2)=0.54 \\ \\ c=0.54(c+2) \\ c-0.54c=2(0.54) \\ 0.46c=1.08 \\ c=(1.08)/(0.46)=2.35 \end{gathered}`