The question asks whether Rolle's Theorem can be applied to the function $f(x) = -x^2 + 30x$ on the closed interval $[0, 30]$, and if it can, to find all values of $c$ in the open interval $(0, 30)$ such that $f'(c) = 0$.

Step 1: Conditions for Rolle's Theorem

Rolle's Theorem can be applied to a function $f(x)$ on a closed interval $[a, b]$ if the following conditions are met:

1. Continuity: The function must be continuous on the closed interval $[a, b]$.
2. Differentiability: The function must be differentiable on the open interval $(a, b)$.
3. Equal values at endpoints: The function must satisfy $f(a) = f(b)$.

Let's check these conditions:

• Continuity: The function $f(x) = -x^2 + 30x$ is a polynomial, and polynomials are continuous everywhere, so it is continuous on $[0, 30]$.

• Differentiability: Polynomials are also differentiable everywhere, so $f(x)$ is differentiable on the open interval $(0, 30)$.

• Equal values at endpoints: $f(0) = -(0)^2 + 30(0) = 0$ $f(30) = -(30)^2 + 30(30) = -900 + 900 = 0$ Since $f(0) = f(30) = 0$, the third condition is satisfied.

Thus, all conditions for Rolle's Theorem are satisfied, so it can be applied.

Step 2: Finding the values of $c$

Now, we need to find $c \in (0, 30)$ such that $f'(c) = 0$.

1. First, find the derivative of $f(x)$: $f'(x) = \frac{d}{dx} \left( -x^2 + 30x \right) = -2x + 30$

2. Set $f'(x) = 0$ to solve for $c$: $-2c + 30 = 0$ $2c = 30$ $c = 15$

Thus, the value of $c$ in the open interval $(0, 30)$ that satisfies $f'(c) = 0$ is $c = 15$.

Conclusion:

• Rolle's Theorem can be applied.
• The value of $c$ in the interval $(0, 30)$ such that $f'(c) = 0$ is $c = 15$.

Would you like any more details or have questions?

Here are some related questions for further practice:

1. How do you find the derivative of higher-degree polynomials?
2. Can Rolle’s Theorem be applied to the function $f(x) = \sin(x)$ on the interval $[0, \pi]$?
3. How does Rolle’s Theorem relate to the Mean Value Theorem?
4. What are the necessary conditions for applying the Mean Value Theorem?
5. How do you apply Rolle’s Theorem to functions with non-polynomial expressions?

Tip: When checking if Rolle's Theorem applies, always verify that the function is continuous and differentiable in the given interval!