To find the middle point $c$ of the function $f(x) = 25 - x^2$ on the interval $[0,4]$, we can use the mean value theorem (MVT) for integrals.

The mean value theorem states that there exists a point $c$ in the interval $[a, b]$ such that:

$f(c) = \frac{1}{b - a} \int_a^b f(x) \, dx$

In this case, we have $a = 0$, $b = 4$, and the function $f(x) = 25 - x^2$. Let's follow these steps:

Step 1: Calculate the integral of $f(x)$ on the interval $[0, 4]$

We need to compute:

$\int_0^4 (25 - x^2) \, dx$

Step 2: Find the value of the integral

$\int_0^4 25 \, dx - \int_0^4 x^2 \, dx$

• The first part is straightforward: $\int_0^4 25 \, dx = 25x \Big|_0^4 = 25(4) - 25(0) = 100$

• The second part is: $\int_0^4 x^2 \, dx = \frac{x^3}{3} \Big|_0^4 = \frac{4^3}{3} - \frac{0^3}{3} = \frac{64}{3}$

Now combine the two results:

$\int_0^4 (25 - x^2) \, dx = 100 - \frac{64}{3} = \frac{300}{3} - \frac{64}{3} = \frac{236}{3}$

Step 3: Apply the mean value theorem

Now use the mean value theorem formula:

$f(c) = \frac{1}{4 - 0} \cdot \frac{236}{3} = \frac{236}{12} = \frac{59}{3}$

So, we want to solve for $c$ where:

$f(c) = 25 - c^2 = \frac{59}{3}$

Step 4: Solve for $c$

Set $25 - c^2 = \frac{59}{3}$:

$25 - \frac{59}{3} = c^2$

First, get a common denominator:

$25 = \frac{75}{3}, \quad \frac{75}{3} - \frac{59}{3} = c^2$

$\frac{75 - 59}{3} = c^2$

$\frac{16}{3} = c^2$

Finally, solve for $c$:

$c = \sqrt{\frac{16}{3}} = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3}$

Thus, the middle point $c$ is:

$c = \frac{4\sqrt{3}}{3}$

Would you like more details on any step? Here are some additional related questions:

1. What is the interpretation of the mean value theorem in calculus?
2. How does the integral of a function relate to its area under the curve?
3. What is the significance of a middle point in a mathematical context?
4. How do you compute definite integrals with more complex functions?
5. What is the geometric interpretation of solving for $c$ in the mean value theorem?

Tip: Practice solving for middle points and applying the mean value theorem with different types of functions to strengthen your understanding of the concept.