The problem asks to determine the average value of the function $f(x) = \frac{2}{5}x^4$ over the interval $[-1, 1]$.

1. Problem Description

The problem asks to determine the average value of the function $f(x) = \frac{2}{5}x^4$ over the interval $[-1, 1]$.

2. Solution Steps

The average value of a function $f(x)$ over an interval $[a, b]$ is given by:

$f_{avg} = \frac{1}{b-a}\int_{a}^{b} f(x) dx$

In this case, $f(x) = \frac{2}{5}x^4$, $a = -1$, and $b = 1$. Substituting these values into the formula, we get:

$f_{avg} = \frac{1}{1 - (-1)}\int_{-1}^{1} \frac{2}{5}x^4 dx$

$f_{avg} = \frac{1}{2} \int_{-1}^{1} \frac{2}{5}x^4 dx$

$f_{avg} = \frac{1}{2} \cdot \frac{2}{5} \int_{-1}^{1} x^4 dx$

$f_{avg} = \frac{1}{5} \int_{-1}^{1} x^4 dx$

Now we evaluate the integral:

$\int x^4 dx = \frac{x^5}{5}$

So,

$\int_{-1}^{1} x^4 dx = \frac{x^5}{5}\Big|_{-1}^{1} = \frac{(1)^5}{5} - \frac{(-1)^5}{5} = \frac{1}{5} - \frac{-1}{5} = \frac{1}{5} + \frac{1}{5} = \frac{2}{5}$

Substitute the result of the integral back into the expression for $f_{avg}$:

$f_{avg} = \frac{1}{5} \cdot \frac{2}{5} = \frac{2}{25}$

3. Final Answer

The average value of the function $f(x) = \frac{2}{5}x^4$ over the interval $[-1, 1]$ is $\frac{2}{25}$.