Let's go step by step to address the questions from your image.

The given revenue function is:

$R(x) = x \cdot p = 1764x - 0.12x^3$

To solve parts A and B, we'll need to calculate the first and second derivatives of the revenue function, as these are used to determine local minima and concavity.

Step 1: First derivative (R'(x))

The first derivative helps to find the critical points:

$R'(x) = \frac{d}{dx} \left(1764x - 0.12x^3 \right) = 1764 - 3(0.12)x^2$

Simplify:

$R'(x) = 1764 - 0.36x^2$

Set the first derivative equal to zero to find the critical points:

$1764 - 0.36x^2 = 0$

Solving for $x$:

$0.36x^2 = 1764 \quad \Rightarrow \quad x^2 = \frac{1764}{0.36} = 4900$ $x = \sqrt{4900} = 70$

So, $x = 70$ is a critical point. This is where the function may have a local minimum or maximum.

Step 2: Second derivative (R''(x))

Now, we calculate the second derivative to find where the graph is concave up or concave down:

$R''(x) = \frac{d}{dx}(1764 - 0.36x^2) = -0.72x$

To find concavity, we analyze the sign of $R''(x)$:

• If $R''(x) > 0$, the graph is concave upward.
• If $R''(x) < 0$, the graph is concave downward.

At $x = 70$, $R''(70) = -0.72(70) = -50.4$, which is negative, indicating concave downward at $x = 70$.

To determine when the graph is concave upward, solve $R''(x) > 0$:

$-0.72x > 0 \quad \Rightarrow \quad x < 0$

However, since $x$ must be between 0 and 95 (given the problem's constraints), there is no interval where the graph is concave upward.

Conclusion:

• Part A: The local minimum occurs at $x = 70$.
• Part B: The graph is never concave upward.

Would you like further clarification on any of these steps?

Here are five related questions:

1. How do you confirm if $x = 70$ is a local minimum or maximum?
2. What is the significance of the second derivative in determining concavity?
3. Can a revenue function like this have multiple critical points?
4. How would the solution change if the revenue function had a different cubic coefficient?
5. Why is concavity important in understanding revenue models?

Tip: Always check the signs of the first and second derivatives to understand the behavior of functions, especially when dealing with maxima, minima, and concavity.