Solution of Business Mathematics

Steps and Hints: Exercise 2 (Q. 1-20)

1. Stationary Points and Inflection (Q1, Q9, Q10)

The Concept:

• Step 1: Find the first derivative (dy/dx). Set it to 0 to find Stationary Points.
• Step 2: Find the second derivative (d²y/dx²). Set it to 0 to find the Point of Inflection.
• Hint: If the second derivative is a constant (like 2 or 6), there is no point of inflection because it can never be zero.

2. Increasing and Decreasing Functions (Q2, Q3)

The Concept:

• Step 1: Differentiate the function.
• Step 2: Plug in the given x-value.
  • Result is Positive (+) = Increasing.
  • Result is Negative (-) = Decreasing.
• Hint for Q3: To find intervals, find the critical points first, then test a number from each "section" of the number line.

3. Elasticity of Demand (Q4, Q5, Q6, Q7)

The Concept:

• Step 1: Identify P (Price) and Q (Quantity).
• Step 2: Find the derivative dQ/dP.
• Step 3: Use the formula: Elasticity = - (P / Q) * (dQ / dP).
• Hint for Q7: First, set Demand = Supply to find the "Equilibrium" P and Q, then calculate elasticity.

4. Maximum and Minimum Values (Q8, Q11, Q12)

The Concept:

• Step 1: Find the first derivative and set to 0 to find critical points.
• Step 2: For a closed interval [a, b], test the critical points AND the endpoints (a and b).
• Hint for Q11/12: Turn the word problem into a single equation first. For example, if Sum = 10, the numbers are x and (10-x).

5. Profit Maximization (Q13 - Q20)

The Concept:

• Step 1: Find Revenue (R = Price * Quantity).
• Step 2: Find Profit (Profit = Revenue - Cost).
• Step 3: Differentiate the Profit equation and set to 0.
• Hint for Q19/20: Profit is maximum when Marginal Revenue (MR) = Marginal Cost (MC). You can differentiate R and C separately and set them equal to each other!