Question

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

Answer 1

Given:

`f(x)=3x^3,\text{ on the interval }\lbrack0,2\rbrack`

First, find the average value of the function in the given interval.

`\begin{gathered} f_(avg)=(1)/(2-0)\int ^2_03x^3dx \\ =(1)/(2)\lbrack(3x^4)/(4)\rbrack^2_0 \\ =(1)/(2)((3)/(4)\cdot2^4-0) \\ =6 \end{gathered}`

Now,

`\begin{gathered} \text{Solve, 3x}^3=6 \\ x^3=(6)/(3) \\ x^3=2 \\ \text{Take cube root} \\ x=\sqrt[3]{2}\in\lbrack0,2\rbrack \end{gathered}`

Answer:

`c=\sqrt[3]{2}`