Let's solve the problem step by step.

Part (b):

Given:

Revenue function: $R(x) = 100x - 0.2x^2$
Cost function: $C(x) = 5000 + 20x$

i) The profit function $P(x)$:

Profit function is defined as:
$P(x) = R(x) - C(x)$
Substituting the given functions:
$P(x) = (100x - 0.2x^2) - (5000 + 20x)$ $P(x) = 100x - 0.2x^2 - 5000 - 20x$ $P(x) = -0.2x^2 + 80x - 5000$

ii) The marginal profit function $P'(x)$:

The marginal profit is the derivative of $P(x)$:
$P'(x) = \frac{d}{dx}(-0.2x^2 + 80x - 5000)$ $P'(x) = -0.4x + 80$

iii) The level of demand $x$ which maximizes the firm's profit:

To find the maximum, set $P'(x) = 0$:
$-0.4x + 80 = 0$ $0.4x = 80$ $x = 200$

iv) Determine the firm's profit at $x = 200$:

Substitute $x = 200$ into $P(x)$:
$P(200) = -0.2(200)^2 + 80(200) - 5000$ $P(200) = -0.2(40000) + 16000 - 5000$ $P(200) = -8000 + 16000 - 5000$ $P(200) = 3000$

v) Verify that the level of demand $x = 200$ is indeed a maximum:

To verify, check the second derivative of $P(x)$:
$P''(x) = \frac{d}{dx}(-0.4x + 80)$ $P''(x) = -0.4$ Since $P''(x) < 0$, the profit function is concave down, and $x = 200$ is indeed a maximum.

Part (c):

Find and classify the stationary points of $f(x) = x^3 - 3x^2 + 6$.

Step 1: Find the derivative $f'(x)$:

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

Step 2: Find the stationary points by setting $f'(x) = 0$:

$3x^2 - 6x = 0$ Factorize:
$3x(x - 2) = 0$ $x = 0 \quad \text{or} \quad x = 2$

Step 3: Classify the stationary points using the second derivative:

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

Evaluate $f''(x)$ at each stationary point:

• At $x = 0$:
  $f''(0) = 6(0) - 6 = -6$ Since $f''(0) < 0$, $x = 0$ is a local maximum.

• At $x = 2$:
  $f''(2) = 6(2) - 6 = 6$ Since $f''(2) > 0$, $x = 2$ is a local minimum.

Stationary points:

• Local maximum at $x = 0$ with $f(0) = 6$
• Local minimum at $x = 2$ with $f(2) = 2^3 - 3(2)^2 + 6 = 8 - 12 + 6 = 2$

Let me know if you have questions about these steps or need further details! Here are 5 related questions:

1. How would you interpret marginal profit in real-world terms?
2. Can you extend the process to find the break-even point where profit is zero?
3. How can we confirm global maxima or minima for $f(x)$?
4. What is the economic significance of $P''(x)$?
5. How do the stationary points of $f(x)$ relate to its graph?

Tip: Always double-check your derivatives to ensure accuracy when solving optimization problems!