Self-Test 05 - Single Variable Optimization

Check your understanding

This self-test lets you check your understanding of Session 05. It is purely formative: nothing is tracked or graded, and nobody sees your answers. Work through the Tasks first, then use this quiz to confirm the ideas have stuck. A wrong answer triggers a 10-second cooldown before you can retry, so think before you click.

1. A stationary point of \(f\) is a point where:

2. Let \(f(x) = x^2 - 6x + 1\). At which \(x\) is its minimum? Type the number.

3. The function \(f(x) = x^3 - 12x\) has a local minimum at a positive \(x\). Type that \(x\).

Classifying with the Second Derivative

4. At a stationary point \(x^*\) you find \(f''(x^*) > 0\). The point is a local:

5. At \(x = 4\), \(f''\) changes sign from negative to positive, and \(f'(4) \neq 0\). The point at \(x = 4\) is:

6. A point has \(f'(x^*) = 0\) and \(f''(x^*) = 0\). What can you conclude?

Local, Global & Boundaries

7. A smooth function has a local maximum at \(x = 2\) on the unbounded domain \(\mathbb{R}\). Is it necessarily the global maximum?

8. To find the global maximum of a continuous \(f\) on a closed interval \([a, b]\), you must compare:

Optimization in Business

9. For an interior optimum, profit \(\pi(q) = R(q) - C(q)\) is maximized where:

10. Profit is \(\pi(q) = -q^2 + 40q - 100\). At which \(q\) is profit maximized? Type the number.

11. In the EOQ formula \(q^* = \sqrt{\tfrac{2DK}{h}}\), the holding cost \(h\) rises while \(D\) and \(K\) stay fixed. The economic order quantity \(q^*\):

12. A warehouse has annual demand \(D = 800\), order cost \(K = 25\) and holding cost \(h = 4\) (both in €). Compute the economic order quantity \(q^* = \sqrt{2DK/h}\). Type the number.

Well done. You solved every question. You can find stationary points with the FOC, classify them with the SOC, spot a saddle when the test falls silent, compare interior points against boundaries, and apply the profit rule and the EOQ. You are ready for Session 06 on systems of linear equations.