Find the average value of the function $f(x)=2 x^{4}$ on the interval $1 \leq x \leq 5$
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Step 1 ：The average value of a function \(f(x)\) on the interval \(a \leq x \leq b\) is given by the formula \(\frac{1}{b-a} \int_{a}^{b} f(x) dx\).

Step 2 ：In this case, we need to find the average value of the function \(f(x)=2x^{4}\) on the interval \(1 \leq x \leq 5\).

Step 3 ：So, we need to calculate the integral of \(f(x)\) from 1 to 5, and then divide it by the length of the interval, which is \(5-1=4\).

Step 4 ：The average value of the function \(f(x)=2x^{4}\) on the interval \(1 \leq x \leq 5\) is \(\frac{1562}{5}\).

Step 5 ：Final Answer: The average value of the function \(f(x)=2x^{4}\) on the interval \(1 \leq x \leq 5\) is \(\boxed{\frac{1562}{5}}\).