(a) Find the average value of $f(x)=25-x^{2}$ on the interval [0,4]. Answer: (b) Find a value $c$ in the interval $[0,4]$ such that $f(c)$ is equal to the average value. Answer:

Step 1 ：The average value of a function \(f(x)\) on the interval \([a,b]\) is given by the formula: \[\frac{1}{b-a}\int_{a}^{b}f(x)dx\]

Step 2 ：In this case, \(f(x) = 25 - x^{2}\), \(a = 0\), and \(b = 4\). So we need to calculate the integral of \(f(x)\) from \(0\) to \(4\) and then divide by \(4-0\).

Step 3 ：The average value of the function \(f(x) = 25 - x^{2}\) on the interval \([0,4]\) is \(\frac{59}{3}\).

Step 4 ：Now, we need to find a value \(c\) in the interval \([0,4]\) such that \(f(c)\) is equal to the average value. This means we need to solve the equation \(f(c) = \frac{59}{3}\).

Step 5 ：A value \(c\) in the interval \([0,4]\) such that \(f(c)\) is equal to the average value is \(\frac{4\sqrt{3}}{3}\).

Step 6 ：Final Answer: The average value of the function \(f(x) = 25 - x^{2}\) on the interval \([0,4]\) is \(\boxed{\frac{59}{3}}\). A value \(c\) in the interval \([0,4]\) such that \(f(c)\) is equal to the average value is \(\boxed{\frac{4\sqrt{3}}{3}}\).