Training Optimization Placement Test Practice — Optimization
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Placement Test Practice — Optimization

24 min Optimization

Placement Test Practice — Optimization

Practice Test — 20 Questions

1. Critical points of $f(x) = x^3 - 12x$.
2. Classify: local max, local min for Problem 1.
3. Maximize area of rectangle with perimeter 24.
4. Maximize $z = 2x + 3y$: $x + y \leq 6$, $x \leq 4$, $x,y \geq 0$.
5. Lagrange: maximize $xy$ with $x + y = 8$.
6. Absolute min of $f(x) = x^2$ on $[-3, 5]$.
7. Farmer: 300 m fence, 3 sides of rectangle against river. Max area?
8. Is $(0,0)$ a corner of $x + y \leq 5$, $x,y \geq 0$?
9. What is a constraint?
10. Minimize $f = x + 9/x$, $x > 0$.
11. LP: min $z = x + y$: $x + 2y \geq 4$, $x,y \geq 0$. Answer?
12. What does $\nabla f = \lambda \nabla g$ mean geometrically?
13. Inflection of $f(x) = x^3 - 3x^2$?
14. Can an LP have infinitely many solutions?
15. Two positive numbers: product 36. Minimize sum.
16. Lagrange: minimize $x^2+y^2$ with $x+y=2$.
17. What is the feasible region for $x \geq 0$, $y \geq 0$?
18. $f(x) = -x^4 + 2x^2$: critical points?
19. Extreme Value Theorem requires what?
20. Revenue $R = 50x - 0.5x^2$. Max revenue?
Show Answer Key

1. $f'=3x^2-12=0$; $x=\pm 2$

2. $f''=6x$. At $x=-2$: $-12<0$ (max). At $x=2$: $12>0$ (min).

3. Square $6 \times 6$; $A = 36$

4. Corners: $(0,0),(4,0),(4,2),(0,6)$. $z(0,6)=18$. Max $=18$.

5. $x = y = 4$. $f = 16$.

6. $f(0) = 0$

7. $A = x(300-2x)$; $x = 75$; $A = 11{,}250\text{ m}^2$

8. Yes

9. A condition that restricts the set of allowable solutions

10. $x = 3$; $f(3) = 6$

11. Corners: $(4,0)$: $z=4$; $(0,2)$: $z=2$. Min $= 2$.

12. Gradient of $f$ is parallel to gradient of $g$; level curves tangent

13. $f''=6x-6=0$; $x=1$. Inflection at $(1,-2)$.

14. Yes — when objective is parallel to a binding constraint edge

15. $x = y = 6$; sum $= 12$

16. $x = y = 1$; $f = 2$

17. First quadrant (including axes)

18. $f'=-4x^3+4x=0$; $x=0, \pm 1$

19. $f$ continuous on a closed, bounded interval

20. $R'=50-x=0$; $x=50$. $R=1{,}250$.

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