Prova de PNL - IME-USPwalterfm/cursos/mac0427/2008/pnlNoturno1.pdf · Prova de PNL Aluno: No USP RG Prova 2 Aviso: as provas contem quest´ oes parecidas mas com n˜ umeros diferentes

Prova de PNL

Walter Mascarenhas

30 de abril de 2008

Prova de PNL

Aluno: No USP RG Prova 1

Aviso: as provas contem questoes parecidas mas com numeros diferentes. Parafacilitar a correcao, DEVOLVA ESSA FOLHA e ESCREVA O NUMERO DAPROVA nas suas folhas de resposta.

Questao 1 (2,0 pontos) Desenhe o grafico da funcao f abaixo, destacando seusmaximos e mınimos locais e os pontos nos quais f nao tem derivada.

f (x) = min{

4x2−6x−4, 4x2 +12x+5}

.

Questao 2 (2,0 pontos) Considere as funcoes

f1(x,y) = −27x2 +27x−27y2−54y−31, (1)f2(x,y) = −27x2−54x−27y2−54y−58. (2)

Determine o conjunto C dos pontos (x,y) para os quais a funcao

f (x,y) = min{ f1(x,y), f2(x,y)}

nao tem gradinte. Desenhe C.

Questao 3 (3,0 pontos) Encontre os maximos e mınimos locais da funcao f daquestao 2. Ha maximos ou mınimos globais? Se houver, quais sao eles?

Questao 4 (3,0 pontos) Desenhe as curvas de nıvel da funcao f da questao 2.

Prova de PNL




f (x) = min{

8x2−4x−8, 8x2−40x+46}

.


f1(x,y) = 1x2−3x+1y2 +2, (3)f2(x,y) = 1x2−12x+1y2 +38. (4)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2 +18x+19, 4x2 +1}

.


f1(x,y) = 2x2 +22x+2y2−8y+60, (5)f2(x,y) = 2x2 +4x+2y2−8y+6. (6)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +2x−4, 2x2−16x+32}

.


f1(x,y) = −2x2 +6x−2y2−4y−3, (7)f2(x,y) = −2x2 +24x−2y2−4y−75. (8)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2−22x+52, 2x2−4x−2}

.


f1(x,y) = 9x2−3x+9y2 +24y+11, (9)f2(x,y) = 9x2−30x+9y2 +24y+38. (10)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2−10x+5, 4x2 +8x+5}

.


f1(x,y) = −9x2−3x−9y2 +12y−1, (11)f2(x,y) = −9x2−30x−9y2 +12y−28. (12)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2−6x, 18x2−60x+54}

.


f1(x,y) = −27x2 +27x−27y2−36y−10, (13)f2(x,y) = −27x2 +108x−27y2−36y−118. (14)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−1x−1, 1x2−10x+26}

.


f1(x,y) = 18x2 +6x+18y2 +48y+26, (15)f2(x,y) = 18x2−48x+18y2 +48y+62. (16)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

12x2 +18x−4, 12x2−36x+23}

.


f1(x,y) = 1x2 +7x+1y2 +7, (17)f2(x,y) = 1x2−2x+1y2−2. (18)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−13x+37, 1x2−4x+1}

.


f1(x,y) = 8x2 +12x+8y2 +24y+22, (19)f2(x,y) = 8x2−24x+8y2 +24y+40. (20)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +52x+84, 8x2 +16x+12}

.


f1(x,y) = −8x2 +52x−8y2−24y−94, (21)f2(x,y) = −8x2 +16x−8y2−24y−22. (22)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2−3x−6, 3x2−30x+75}

.


f1(x,y) = 4x2 +2x+4y2 +16y+10, (23)f2(x,y) = 4x2−16x+4y2 +16y+28. (24)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2 +18x+4, 18x2−36x+22}

.


f1(x,y) = 1x2 +1x+1y2 +8y+16, (25)f2(x,y) = 1x2−8x+1y2 +8y+34. (26)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

27x2 +99x+86, 27x2 +18x+5}

.


f1(x,y) = 4x2−2x+4y2 +4y+1, (27)f2(x,y) = 4x2−20x+4y2 +4y+28. (28)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2 +2x−5, 4x2 +20x+22}

.


f1(x,y) = 27x2 +27x+27y2 +18y+5, (29)f2(x,y) = 27x2 +108x+27y2 +18y+113. (30)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

12x2−18x+4, 12x2−72x+112}

.


f1(x,y) = 27x2 +9x+27y2−10, (31)f2(x,y) = 27x2−72x+27y2 +44. (32)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2 +6x−2, 18x2−48x+34}

.


f1(x,y) = 18x2−78x+18y2 +36y+96, (33)f2(x,y) = 18x2−24x+18y2 +36y+24. (34)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

12x2−42x+32, 12x2 +12x+5}

.


f1(x,y) = 2x2−14x+2y2−4y+24, (35)f2(x,y) = 2x2 +4x+2y2−4y+6. (36)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2 +7x+13, 1x2−2x+4}

.


f1(x,y) = −27x2 +9x−27y2 +8, (37)f2(x,y) = −27x2−72x−27y2−46. (38)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2−54x+35, 18x2−1}

.


f1(x,y) = −27x2−81x−27y2 +54y−79, (39)f2(x,y) = −27x2−27y2 +54y−25. (40)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2−78x+78, 18x2−24x+6}

.


f1(x,y) = 2x2−2x+2y2−8y+8, (41)f2(x,y) = 2x2−20x+2y2−8y+62. (42)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

27x2 +63x+34, 27x2−18x+7}

.


f1(x,y) = 1x2−3x+1y2−6y+11, (43)f2(x,y) = 1x2−12x+1y2−6y+47. (44)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

27x2−9x−9, 27x2 +72x+45}

.


f1(x,y) = 9x2 +3x+9y2−18y+9, (45)f2(x,y) = 9x2 +30x+9y2−18y+36. (46)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +14x+22, 2x2−4x+4}

.


f1(x,y) = 27x2 +27x+27y2−72y+48, (47)f2(x,y) = 27x2−54x+27y2−72y+75. (48)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +44x+54, 8x2 +8x}

.


f1(x,y) = −18x2 +6x−18y2−12y+6, (49)f2(x,y) = −18x2 +60x−18y2−12y−48. (50)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2−9x−4, 3x2−36x+104}

.


f1(x,y) = 4x2 +18x+4y2 +19, (51)f2(x,y) = 4x2 +4y2 +1. (52)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2 +3x+2, 9x2 +30x+29}

.


f1(x,y) = 18x2 +18x+18y2 +24y+6, (53)f2(x,y) = 18x2−36x+18y2 +24y+24. (54)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2 +39x+118, 3x2 +12x+10}

.


f1(x,y) = 1x2 +3x+1y2 +4, (55)f2(x,y) = 1x2 +12x+1y2 +40. (56)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +12x+2, 8x2 +48x+74}

.


f1(x,y) = 1x2 +13x+1y2 +8y+56, (57)f2(x,y) = 1x2 +4x+1y2 +8y+20. (58)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2 +27x+51, 3x2−3}

.


f1(x,y) = −2x2−14x−2y2 +4y−18, (59)f2(x,y) = −2x2 +4x−2y2 +4y. (60)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +6x+1, 2x2 +24x+73}

.


f1(x,y) = −27x2−81x−27y2−72y−104, (61)f2(x,y) = −27x2−27y2−72y−50. (62)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2−9x+3, 9x2−36x+39}

.


f1(x,y) = −18x2 +6x−18y2 +3, (63)f2(x,y) = −18x2−48x−18y2−33. (64)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2 +39x+41, 9x2 +12x+5}

.


f1(x,y) = −27x2−27x−27y2 +18y−7, (65)f2(x,y) = −27x2 +54x−27y2 +18y−34. (66)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +36x+37, 8x2 +1}

.


f1(x,y) = 18x2−78x+18y2−12y+80, (67)f2(x,y) = 18x2−24x+18y2−12y+8. (68)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2 +9x−2, 3x2−18x+25}

.


f1(x,y) = 27x2−9x+27y2 +18y−7, (69)f2(x,y) = 27x2−90x+27y2 +18y+74. (70)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2 +66x+52, 18x2 +12x−2}

.


f1(x,y) = −12x2 +6x−12y2 +36y−20, (71)f2(x,y) = −12x2−48x−12y2 +36y−74. (72)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2 +6x−2, 18x2−48x+34}

.


f1(x,y) = 18x2 +6x+18y2 +12y−3, (73)f2(x,y) = 18x2−48x+18y2 +12y+33. (74)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +26x+82, 2x2 +8x+10}

.


f1(x,y) = 3x2−9x+3y2 +12y+14, (75)f2(x,y) = 3x2−36x+3y2 +12y+122. (76)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2 +13x+39, 1x2 +4x+3}

.


f1(x,y) = 12x2−6x+12y2−24y+3, (77)f2(x,y) = 12x2 +48x+12y2−24y+57. (78)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

27x2 +81x+52, 27x2−2}

.


f1(x,y) = −27x2−27x−27y2 +1, (79)f2(x,y) = −27x2 +54x−27y2−26. (80)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +2x−6, 2x2 +20x+48}

.


f1(x,y) = −18x2−66x−18y2 +48y−91, (81)f2(x,y) = −18x2−12x−18y2 +48y−37. (82)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +12x, 8x2 +48x+72}

.


f1(x,y) = −3x2 +39x−3y2−18y−149, (83)f2(x,y) = −3x2 +12x−3y2−18y−41. (84)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2−9x−1, 9x2−36x+35}

.


f1(x,y) = −12x2 +78x−12y2−12y−125, (85)f2(x,y) = −12x2 +24x−12y2−12y−17. (86)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +26x+78, 2x2 +8x+6}

.


f1(x,y) = 3x2−21x+3y2−24y+80, (87)f2(x,y) = 3x2 +6x+3y2−24y+53. (88)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2 +6x+2, 4x2 +24x+38}

.


f1(x,y) = 12x2−42x+12y2−24y+45, (89)f2(x,y) = 12x2 +12x+12y2−24y+18. (90)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2 +21x+14, 9x2−6x+5}

.


f1(x,y) = 1x2 +1x+1y2−8y+14, (91)f2(x,y) = 1x2−8x+1y2−8y+32. (92)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2−14x+13, 4x2 +4x+4}

.


f1(x,y) = −12x2−18x−12y2−48y−45, (93)f2(x,y) = −12x2−72x−12y2−48y−153. (94)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2 +3x−7, 3x2−24x+47}

.


f1(x,y) = 8x2 +12x+8y2 +24y+20, (95)f2(x,y) = 8x2−24x+8y2 +24y+38. (96)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2 +27x+50, 3x2−4}

.


f1(x,y) = −27x2−45x−27y2−18y−16, (97)f2(x,y) = −27x2 +36x−27y2−18y−16. (98)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−7x+14, 1x2 +2x+5}

.


f1(x,y) = 1x2−11x+1y2 +8y+44, (99)f2(x,y) = 1x2−2x+1y2 +8y+17. (100)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2 +54x+33, 18x2−3}

.


f1(x,y) = −9x2−27x−9y2 +24y−36, (101)f2(x,y) = −9x2−9y2 +24y−18. (102)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2 +21x+10, 9x2−6x+1}

.


f1(x,y) = 2x2 +26x+2y2−12y+98, (103)f2(x,y) = 2x2 +8x+2y2−12y+26. (104)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

27x2−99x+81, 27x2−18x}

.


f1(x,y) = −9x2 +9x−9y2 +6y+1, (105)f2(x,y) = −9x2 +36x−9y2 +6y−35. (106)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2 +2x+2, 4x2 +20x+29}

.


f1(x,y) = −18x2 +18x−18y2 +24y−12, (107)f2(x,y) = −18x2 +72x−18y2 +24y−84. (108)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2 +9x−4, 9x2−18x+5}

.


f1(x,y) = −18x2 +30x−18y2−48y−44, (109)f2(x,y) = −18x2−24x−18y2−48y−44. (110)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2−18x−3, 18x2 +36x+15}

.


f1(x,y) = 3x2 +9x+3y2 +18y+24, (111)f2(x,y) = 3x2−18x+3y2 +18y+51. (112)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2−27x+58, 3x2 +4}

.


f1(x,y) = 4x2−26x+4y2−8y+47, (113)f2(x,y) = 4x2−8x+4y2−8y+11. (114)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2 +14x+13, 4x2−4x+4}

.


f1(x,y) = −12x2 +18x−12y2 +12y−5, (115)f2(x,y) = −12x2−36x−12y2 +12y−32. (116)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2−10x+12, 2x2 +8x+12}

.


f1(x,y) = 9x2−15x+9y2 +12y+9, (117)f2(x,y) = 9x2 +12x+9y2 +12y+9. (118)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−3x+1, 1x2−12x+37}

.


f1(x,y) = 9x2−15x+9y2−24y+20, (119)f2(x,y) = 9x2 +12x+9y2−24y+20. (120)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2−22x+24, 4x2−4x−3}

.


f1(x,y) = −2x2 +10x−2y2 +16y−38, (121)f2(x,y) = −2x2−8x−2y2 +16y−38. (122)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

12x2 +18x+2, 12x2−36x+29}

.


f1(x,y) = −4x2−18x−4y2 +16y−38, (123)f2(x,y) = −4x2−4y2 +16y−20. (124)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2−22x+56, 2x2−4x+2}

.


f1(x,y) = 18x2−18x+18y2 +12y+5, (125)f2(x,y) = 18x2 +36x+18y2 +12y+23. (126)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +52x+76, 8x2 +16x+4}

.


f1(x,y) = 9x2 +33x+9y2 +31, (127)f2(x,y) = 9x2 +6x+9y2 +4. (128)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2−9x−1, 3x2−36x+107}

.


f1(x,y) = 3x2 +15x+3y2−24y+58, (129)f2(x,y) = 3x2−12x+3y2−24y+58. (130)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2 +6x−8, 18x2−48x+28}

.


f1(x,y) = −1x2−11x−1y2−6y−40, (131)f2(x,y) = −1x2−2x−1y2−6y−13. (132)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2−52x+80, 8x2−16x+8}

.


f1(x,y) = −3x2 +15x−3y2−12y−23, (133)f2(x,y) = −3x2−12x−3y2−12y−23. (134)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +36x+32, 8x2−4}

.


f1(x,y) = 1x2−9x+1y2−4y+18, (135)f2(x,y) = 1x2 +1y2−4y. (136)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−13x+40, 1x2−4x+4}

.


f1(x,y) = −12x2 +6x−12y2 +12y+4, (137)f2(x,y) = −12x2 +60x−12y2 +12y−77. (138)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2−9x−2, 9x2−36x+34}

.


f1(x,y) = 9x2 +33x+9y2 +18y+40, (139)f2(x,y) = 9x2 +6x+9y2 +18y+13. (140)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2−30x+4, 18x2 +24x+4}

.


f1(x,y) = −27x2−63x−27y2 +36y−42, (141)f2(x,y) = −27x2 +18x−27y2 +36y−15. (142)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2 +1x−5, 1x2 +10x+22}

.


f1(x,y) = 1x2−3x+1y2−6y+13, (143)f2(x,y) = 1x2 +6x+1y2−6y+22. (144)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2 +33x+83, 3x2 +6x+2}

.


f1(x,y) = 8x2−4x+8y2−8y−3, (145)f2(x,y) = 8x2−40x+8y2−8y+51. (146)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

8x2 +52x+83, 8x2 +16x+11}

.


f1(x,y) = 18x2 +42x+18y2−48y+56, (147)f2(x,y) = 18x2−12x+18y2−48y+38. (148)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2 +1x−1, 1x2 +10x+26}

.


f1(x,y) = 27x2−9x+27y2−36y+2, (149)f2(x,y) = 27x2−90x+27y2−36y+83. (150)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2−42x+17, 18x2 +12x−1}

.


f1(x,y) = −18x2−6x−18y2 +48y−27, (151)f2(x,y) = −18x2 +48x−18y2 +48y−63. (152)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2−42x+18, 18x2 +12x}

.


f1(x,y) = −9x2−39x−9y2−18y−45, (153)f2(x,y) = −9x2−12x−9y2−18y−9. (154)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2 +15x+4, 9x2−12x+4}

.


f1(x,y) = −4x2−2x−4y2−8y−3, (155)f2(x,y) = −4x2 +16x−4y2−8y−21. (156)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

3x2 +27x+54, 3x2} .


f1(x,y) = −18x2−6x−18y2−48y−24, (157)f2(x,y) = −18x2−60x−18y2−48y−78. (158)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

12x2−18x, 12x2 +36x+27}

.


f1(x,y) = 12x2 +18x+12y2 +36y+31, (159)f2(x,y) = 12x2 +72x+12y2 +36y+139. (160)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

4x2−6x−4, 4x2−24x+32}

.


f1(x,y) = −1x2 +11x−1y2 +8y−41, (161)f2(x,y) = −1x2 +2x−1y2 +8y−14. (162)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−3x+3, 1x2 +6x+12}

.


f1(x,y) = −2x2−26x−2y2 +16y−112, (163)f2(x,y) = −2x2−8x−2y2 +16y−40. (164)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

18x2 +54x+39, 18x2 +3}

.


f1(x,y) = −4x2 +10x−4y2−8y−8, (165)f2(x,y) = −4x2−8x−4y2−8y−8. (166)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−11x+30, 1x2−2x+3}

.


f1(x,y) = −1x2 +7x−1y2−6, (167)f2(x,y) = −1x2−2x−1y2 +3. (168)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

9x2 +21x+14, 9x2−6x+5}

.


f1(x,y) = −8x2 +4x−8y2−32y−29, (169)f2(x,y) = −8x2−32x−8y2−32y−65. (170)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2−22x+57, 2x2−4x+3}

.


f1(x,y) = −1x2 +5x−1y2 +6y−11, (171)f2(x,y) = −1x2−4x−1y2 +6y−11. (172)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

1x2−7x+8, 1x2 +2x−1}

.


f1(x,y) = −27x2 +99x−27y2−88, (173)f2(x,y) = −27x2 +18x−27y2−7. (174)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +18x+32, 2x2−4}

.


f1(x,y) = −4x2−6x−4y2 +4y−1, (175)f2(x,y) = −4x2 +12x−4y2 +4y−10. (176)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

2x2 +14x+17, 2x2−4x−1}

.


f1(x,y) = 2x2−2x+2y2−12y+10, (177)f2(x,y) = 2x2−20x+2y2−12y+64. (178)


f (x,y) = min{ f1(x,y), f2(x,y)}




Prova de PNL




f (x) = min{

12x2 +54x+50, 12x2−4}

.


f1(x,y) = 18x2 +66x+18y2 +58, (179)f2(x,y) = 18x2 +12x+18y2 +4. (180)


f (x,y) = min{ f1(x,y), f2(x,y)}




Documents

Prova de PNL - IME-USPwalterfm/cursos/mac0427/2008/pnlNoturno1.pdf · Prova de PNL Aluno: No USP RG Prova 2 Aviso: as provas contem quest´ oes parecidas mas com n˜ umeros diferentes