Solutions Hints and Tips of Business Mathematics Grade 12 ( NEB)

Core Derivative Problems

Example 4: Finding Turning Points

Question: Find the stationary point and points of inflection of y = x^3 - 3x^2 - 9.

1. Finding Stationary Points:
- First derivative: 3x^2 - 6x.
- Set to zero: 3x(x - 2) = 0. This gives x = 0 and x = 2.
- Tip: These are your critical points where the graph changes direction.

2. Finding Point of Inflection:
- Second derivative: 6x - 6.
- Set to zero: 6x = 6, which gives x = 1.
- Tip: At x = 1, the curve stops bending one way and starts bending the other.

Example 3: Intervals of Growth

Question: Find the intervals for which f(x) = x^4 - 2x^2 is increasing or decreasing.

Steps:
- Derivative: 4x^3 - 4x. Factor it: 4x(x - 1)(x + 1).
- Critical points: -1, 0, and 1.
- Trick: Test a number in each zone.

• If derivative is Negative (-): Decreasing.
• If derivative is Positive (+): Increasing.

- Result: It increases when x is between -1 and 0, or when x is greater than 1.

Business Optimization Derivatives

Example 1: Minimizing Average Cost

Question: If Total Cost C = 1/3 Q^3 - 3Q^2 + 9Q, at what output level is Average Cost minimum?

Steps:
- First, find Average Cost (AC) by dividing C by Q: AC = 1/3 Q^2 - 3Q + 9.
- To find the minimum, take the derivative of AC: 2/3 Q - 3.
- Set to zero: 2/3 Q = 3, which gives Q = 4.5.
- Hint: The second derivative of AC is 2/3 (Positive), which proves this is a minimum.

Condition for Max Profit

Key Concept: Profit is highest when Marginal Revenue (MR) = Marginal Cost (MC).

Trick to Solve:
1. Find Revenue (Price * Quantity).
2. Find MR by differentiating Revenue.
3. Find MC by differentiating Total Cost.
4. Set MR = MC and solve for Q.