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]:

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

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

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

Solve the definite integral:

$\begin{gathered} \int a\cdot f(x)dx=a\int f(x)dx \\ \\ \bar{f}=\frac{1}{2}\times5\times\int ^2_0x^3dx \\ \\ \bar{f}=\frac{5}{2}\int ^2_0x^3dx \\ \\ \\ \int x^ndx=\frac{x^{n+1}}{n+1} \\ \\ \\ \bar{f}=\frac{5}{2}\times(\frac{x^4}{4})^2_0 \end{gathered}$$\begin{gathered} \bar{f}=\frac{5}{2}\times(\frac{2^4}{4}-\frac{0^4}{4}) \\ \\ \bar{f}=\frac{5}{2}\times(\frac{16}{4}-0) \\ \\ \bar{f}=\frac{5}{2}\times(4) \\ \\ \bar{f}=\frac{20}{2} \\ \\ \bar{f}=10 \end{gathered}$

________

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

Solve c:

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