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.

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

Now,

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

Answer:

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