Question

Find every point c in the interval [0, 2] at which f(x) = 5x^3 takes on its average value. Separate multiple answers with a comma. Write the exactanswer. Do not round.

Answer 1

Average value of a function in the interval [ a,b]:

`\bar{f}=(1)/(b-a)\int ^b_af(x)dx`

For the given function in the given interval [0, 2]:

`\begin{gathered} f(x)=5x^3 \\ \\ \bar{f}=(1)/(2-0)\int ^2_05x^3dx \\ \\ \bar{f}=(1)/(2)\int ^2_05x^3dx \end{gathered}`

Solve the definite integral:

`\begin{gathered} \int a\cdot f(x)dx=a\int f(x)dx \\ \\ \bar{f}=(1)/(2)*5*\int ^2_0x^3dx \\ \\ \bar{f}=(5)/(2)\int ^2_0x^3dx \\ \\ \\ \int x^ndx=(x^(n+1))/(n+1) \\ \\ \\ \bar{f}=(5)/(2)*((x^4)/(4))^2_0 \end{gathered}`
`\begin{gathered} \bar{f}=(5)/(2)*((2^4)/(4)-(0^4)/(4)) \\ \\ \bar{f}=(5)/(2)*((16)/(4)-0) \\ \\ \bar{f}=(5)/(2)*(4) \\ \\ \bar{f}=(20)/(2) \\ \\ \bar{f}=10 \end{gathered}`

________

`\begin{gathered} f(c)=\bar{f} \\ \\ f(c)=10 \\ \\ \end{gathered}`

Solve c:

`\begin{gathered} f(c)=5c^3 \\ f(c)=10 \\ \\ \\ 5c^3=10 \\ \\ c^3=(10)/(5) \\ \\ c^3=2 \\ \\ c=\sqrt[3]{2} \end{gathered}`