Let’s analyze and solve the questions from the document.

Given:

The function for power $P(n)$ is modeled as: $f(x) = -2x^3 + 12x^2 + 30x$ where $x$ represents the frequency of rotation in thousands of revolutions per minute (rpm), within the interval $[1, 6]$.

1.1.b. For which value of $x$ does the function $f(x)$ appear to have a maximum?

To determine the maximum:

1. Find the derivative $f'(x)$.
2. Solve $f'(x) = 0$ to locate critical points.
3. Test the critical points within $[1, 6]$ to identify the maximum.

Derivative:

$f'(x) = \frac{d}{dx}(-2x^3 + 12x^2 + 30x)$ $f'(x) = -6x^2 + 24x + 30$

Solve $f'(x) = 0$:

$-6x^2 + 24x + 30 = 0$ Simplify: $x^2 - 4x - 5 = 0$ Factorize: $(x - 5)(x + 1) = 0$ $x = 5 \quad \text{or} \quad x = -1$ Since $x$ must be within the interval $[1, 6]$, only $x = 5$ is valid.

Thus, the maximum appears at $x = 5$.

1.2.a. Expression for the derivative $f'(x)$:

From earlier: $f'(x) = -6x^2 + 24x + 30$

1.2.b. Calculate $f'(5)$:

Substitute $x = 5$ into $f'(x)$: $f'(5) = -6(5)^2 + 24(5) + 30$ $f'(5) = -6(25) + 120 + 30$ $f'(5) = -150 + 120 + 30 = 0$

This confirms $x = 5$ is indeed a critical point.

Would you like me to graph the function or provide more detailed steps? Let me know! Here are five additional questions based on this:

1. How do you confirm whether $x = 5$ is a maximum or a minimum?
2. What is the value of $f(x)$ at $x = 5$?
3. How can the second derivative test be applied to verify the nature of the critical point?
4. What happens to $f(x)$ as $x \to 6$?
5. Can $f(x)$ have another critical point within the interval $[1, 6]$?

Tip:

Always verify critical points by substituting into the second derivative $f''(x)$ or testing intervals around them to confirm maxima or minima.