Derivatives BM NEB 12

Question 1: Stationary Points and Inflection

Question: Find the stationary points and point of inflection of the function y = x^3 - 3x^2 - 9x + 15.

Solution:
1. First Derivative: dy/dx = 3x^2 - 6x - 9.
2. For stationary points, set dy/dx = 0: 3(x^2 - 2x - 3) = 0. Solving gives x = 3 and x = -1.
3. At x = 3, y = -12. At x = -1, y = 20. Stationary points are (3, -12) and (-1, 20).
4. Second Derivative: d2y/dx2 = 6x - 6.
5. For inflection, set d2y/dx2 = 0: 6x = 6, so x = 1.
6. At x = 1, y = 4. Point of inflection is (1, 4).

Tips and Tricks:
- The "Flat" Rule: Stationary points occur where the slope is zero (flat). Always set the first derivative to 0.
- The "Bend" Rule: Inflection points occur where the curve changes its bend. Always set the second derivative to 0.
- Check: Use x = 0 to quickly find the y-intercept if you need to sketch the graph!

Question 3: Increasing and Decreasing Intervals

Question: Determine the intervals in which f(x) = 2x^3 - 15x^2 + 36x + 1 is increasing or decreasing.

Solution:
1. Find f'(x) = 6x^2 - 30x + 36.
2. Set f'(x) = 0 and divide by 6: x^2 - 5x + 6 = 0. Solving gives x = 2 and x = 3.
3. Test intervals: - For x < 2: f'(0) = 36 (Positive). - For 2 < x < 3: f'(2.5) = -1.5 (Negative). - For x > 3: f'(4) = 12 (Positive).
4. Result: Increasing on (-infinity, 2) and (3, infinity). Decreasing on (2, 3).

Hints and Hints:
- Sign Check: If the derivative is positive (+), the function is climbing up. If negative (-), it is sliding down.
- Trick: For most cubic functions, if it starts by increasing, it will alternate: Increasing -> Decreasing -> Increasing.

Question 17: Profit Maximization

Question: Demand function P = 20 - Q and Cost C = Q^2 + 8Q + 2. Find output Q for maximum profit.

Solution:
1. Revenue R = P * Q = (20 - Q)Q = 20Q - Q^2.
2. Profit = R - C = (20Q - Q^2) - (Q^2 + 8Q + 2) = -2Q^2 + 12Q - 2.
3. Max Profit condition: d(Profit)/dQ = 0.
4. -4Q + 12 = 0, which means 4Q = 12. So, Q = 3.
5. At Q = 3, Price P = 20 - 3 = 17.

Business Tips:
- Shortcut: Profit is maximum when Marginal Revenue (MR) = Marginal Cost (MC).
- Check: Total Revenue must always be P times Q. Don't forget to multiply!

Question 20: Cost Minimization vs. Profit

Question: Fixed market price is 500. Total Cost C = 1/3 Q^3 - Q^2 + 452Q + 50. Show max profit at Q = 8.

Solution:
1. Revenue R = 500 * Q.
2. Profit = 500Q - (1/3 Q^3 - Q^2 + 452Q + 50).
3. Derivative of Profit = 500 - Q^2 + 2Q - 452 = -Q^2 + 2Q + 48.
4. Set to zero: Q^2 - 2Q - 48 = 0. Factors are (Q - 8)(Q + 6) = 0.
5. Q = 8 (since quantity cannot be negative).

Hints:
- Fixed Price: In a perfect market, your Price is your Marginal Revenue.
- Factorization Tip: If you struggle with factoring, use the quadratic formula to find Q quickly.