1.

DEMUESTRE QUE:

L ℑ(x , y , z)→(1,4,9)

√ x+√ y+√ z+√xy+√xz+√ yz=17

Demostración:∀ ε>0 ,∃ δ>0 /|f (x , y ,z )−17|<ε Siempre que: 0<√(x−1)2+( y−4)2+(z−9)2<δ

Además:¿(x , y , z )−(1,4,9)∨¿δ⇨∨x−1∨¿δ∧∨ y−4∨¿δ∧∨z−9∨¿δ

Luego:

|√ x+√ y+√z+√ xy+√ xz+√ yz−17|=|√ x+√ y+√ z+ (√x−1 )+√ y+√x (√ z−3 )+3√ x+√z (√ y−2 )+2√z−17|

|√ x+√ y+√z+√ xy+√ xz+√ yz−17|=|4 √x+2√ y+3 √z+√ y (√x−1 )+√x (√z−3 )+√z (√ y−2 )−17|

|√ x+√ y+√z+√ xy+√ xz+√ yz−17|=|4 (√x−1 )+4+2 (√ y−2 )+4+3 (√z−3 )+9+√ y (√x−1 )+√x (√ z−3 )+√ z (√ y−2 )−17|

|√ x+√ y+√z+√ xy+√ xz+√ yz−17|=|4 (√x−1 )+2 (√ y−2 )+3 (√z−3 )+√ y (√x−1 )+√x (√z−3 )+√z (√ y−2 )|

|√ x+√ y+√z+√ xy+√ xz+√ yz−17|=|4 ( x−1 )√ x+1

+2 ( y−4 )√ y+2

+3 (z−9 )√ z+3

+ √ y ( x−1 )√x+1

+ √ x ( z−9 )√z+3

+ √ z ( y−4 )√ y+2 |

|√ x+√ y+√z+√ xy+√ xz+√ yz−17|=4¿

Si asumo f=1 (radio 1 de la esfera)

|x−1|<1−1<x−1<10<x<20<√x<√2…<2|√ x|<√2…①0<√x<√21<√ x+1<√2+1

1

√2+1< 1

√x+1<1

| 1

√ x+1|<1… ②

|y−4|<1−1< y−4<13< y<5√3<√ y<√5…<3|√ y|<3…③√3<√ y<√5√3+2<√ y+2<√5+2

1

√5+2< 1

√ y+2< 1

√3+2<1

| 1

√ y+2|<1…④

|z−9|<1−1<z−9<18<z<10√8<√ z<√10…<4|√ z|<4…⑤√8<√ z<√10√8+3<√z+3<√10+3

1

√10+3< 1

√ z+3< 1

√8+3<1

| 1

√ z+3|<1…⑥ , , ,① ② ③ ④ , , en ⑤ ⑥ *

4 δ+2δ+3δ+3 δ+2δ+4δ<ε

18δ<ε⇨δ< ε18

δ= ε18

Luego:

|√ x+√ y+√z+√ xy+√ xz+√ yz−17|<εSiempre que:

0<√(x−1)2+( y−4)2+(z−9)2<δDónde:

δ=min {1,ε

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