Esercizi d'esame di funzioni in più variabili

NON

get Mat

(1 x) %

l 1

x y

0 =

=

=

-

-/n 1g) &

x

2 =

0 y =

=

+

xe &

E 1

1 X

E =

X = -

e

. +g)

x(e- yy)

0 x(

= e

=

-

2

y 2

y =

= )

2) (e

0) (1

(0 ,

,

, )

Y -ky)

(

fx -

ye -x

xya

= + .

(B) xy)

(y

fx = -

2 . *

** (

B(y x)

xy) i

-

fxx +

x -

-

= - -

xe 0

1 =

by xye

+ -

.

= ** Exy)

/x

-

fy 2

= - e

x y)(x 1xy)

fyy 12 =

-

= - * x2)

y(1

B)(y

( xy) -

fxy - e -

+

12 -

= = (e +ofu)L

-

(0)/82) e

= -

2)

(1 est)

,

H-) e

( 2)

1

- , Leak I

I

↳ d

= =

SluB

Kunzone

I t

f (x g) 2 x

-

-

=

, zy 0

=

-

y5 -

0

x - =

2x

Ax -

-

zy

= - 3)

( =

y5 -zy

2x -

=

-

y -2x

fy y5

-

= y15

x -

= + -

=

zy

- -

2

/*

2)

-

8) 0

= 1"f4 I

= - y y =

-

=

y F

-

+ y =

y =

e

=

y P

-

0

↓ = H

*

x =

= T

2

3x2

fxx byz)

(

= 3x

- 2

-

-sys -

byg M

= = I

~ -

Fyx

-xy 2 =

= - (i Y Sena

)) >

14/0 = - 0

=

, - 3 tri))

)

(

rb) - To

+, =

Ha( = AX

(e) )

frb) To

Ha(-E =

=

, Ax

f(x 3X2

x 3y2x

3y2

y) - -

- +

, = 3y2

3x

Ax 6x +

-

= - 6yx

+

6y

fy = - 3 =

=

3y2

E -

3x2 =0

6x 6x)=0

y(

+

- - 6 +

- - 1

6yx =0

6y + x =

- 0

y =

X 1

= 2

3y2 6x

3x 0

=

-

-

0

=

+

6

3

- - - 2x 0

3y2 x =

9 -

= =

X 0

x( 2) =

x 0

=

- -

2 3 x

y 2

=

= -

r3

I

=

y ) 0)

/

(0 0)

( 2)(1 - 2

-

, ,

+ ,

,

AXX a

6X 6 o

f

= - - -oxs

fyg +

6 6x

= - fyx

fxy by

= = [00)

lol a

360

= s an

(5)

-)

( (n 7340320 sea

+ =

= 63)

(

)

# (1 340320

- f

- - see

=

=

y

x 2y

f(x 2x

y) 3

+

+

= -

-

, 1)

4x(x

E

(axsh +0 v 1

+ 1

x x

-

1

- = y2

4y( 1) yoy + 1

I

+

my - 1)

Pz(1

0)

P3(

1)

P3(0 1 -

d)

Pr(0 1)

, Pq)

- , 1

-

, - ,

, 1)

1) P8(

P6(1

P4(1

1) )

P2(0 1

, - ,

,

, 163

y)

12x2

fx 4

= f(x fx

- y

=

, ,

8 D

=

12y2

byy 4

= +

-

(fitbit) (kx eith

H : - i

)

- O Bet Negato -

)

H(o :

=

, punto di SERA

(

4)

= )

H(ot -

= 30

= =

2+

- MAX

p

5 -

) 3)

H(o - =

, 20 messimo

Punto

>

- 70

DeT

SE VMIN

MAX

08 e

(k )

-

H(1 ) =

, = di

punto

↳ MINIMO

((+3)

H( 0) Mine

32 a

1 =

- =

,

f(x 2y

y3

2

x

y) 4x

+ -

=

, (3x2 (2 e -

8x y

+ x 0 v

x =

=

f

- = y(3y 4) =

vy

y =

-

&2) f)

7910)

P(0 0) /3

"6)

P P

, ,

,

O ,

fxx 6x 8

+

= 6

y) (y)

f(x fx

fyg 6y 4 =

-

= , ,

8 D

=

(6x +

Ex3)

(fir

He = ↓

(6x +

Ex3)

(fir

He =

) ) SellA

To

Po PUN

3 20

-

= = -

) 8 (

80) )

6 +

- .

P Moses

= 32

= so

=

. -

(80

) 8 0 40.) )

E = MIMMO

>

= =

)

( -604 +8 /)

P3 = sechs

to

32 prem

= - <o

:

: 0

det

Se indeterminato W

>

= - I

(

f(x x y 2y2

y) 2x 3

+

= +

, - -

4 4x3 -y 2y2

4x +

-

Of 2x

= 4y3

x4 hy

+

-

- x = 0

f'(x) 4x2 1)

4x(x -

0

=

4x

= -

- - 11

=

x

hy3

f(y) by

+

-

= 4y(y2 1) -

y =

-

0

=

-

- - Is

y =

d)

1)( 0)(

1)/0

(0 +

1 e

- - ,

,

,

, (

( 12x

12x

f"(x) 4

3 - D

-

=

f"(y) 124 3

+

= 124th

- 0 -

f(xy) &

= ) a) 1)

H(01) H(0

- MN

32

= =

+ , -

=

+

n n)

b) ( (1 3

H( 0) - MNx

n

= 32

, =

, +

=

" +

y) bxy

f(x +

2 by

x +

=

, 4x

Ax 4x

+ hy

+

=

by bx

8y +

= 8y Ex

y

4x

By + hx = = -

= 0 -

4f(x)

4x -

4x =

+

+

4x 4x

hx-2x ex =

0

+ = +

1) x

2x(2x = -

-

+ 0

= 11

-

)

(0 I

quindi x

20 = Ve

, "

12x2

fxx 12x 4

h ↳

+

=

fyy E

= ·

4

fxy byx 4

=

= (i

0)

H(0 16

32

= 16

, -

= =

y)

A(x = 15x 12y

x -

3xy

+

=

, -

3y2

x 15

+

3x -

=

by 1

126x =

6yx 6xy x

=

=

12

= -

3(5) 34 by

12

13

+ :

0

=

- + 1542

3y" 159 03y2 12 0

=

+

12 + =

-

y2

w = 30 =

15W 12 30th

+ W

= =

0

- - 4 wi

16

2 -

We -

= =

= -

2 W 1

=

=

y =

4y 2

=

=

y 1y = 1

= + 1

y 2 x

1 X =

y = 2 =

= -

-

y 2

X

1

= y

= = x

2 1

=

- -

Axx 6x

6x by

= 6 x by

Ayy 6x

= 8yx

↓ 6y

=

xy = (* )

H(1 se

36 -144

2) so

=

, =

H( (

2) [

1

- - = seas

36 143<0

= -

(n)

H(41) 14b Mun

3670

-

=

=

H( f

1) )

(2

2 MA

144

- 3620

- , -

= -

=

2

y)

f(x 8xy + 1

2x +

+

=

, 8y3x

1x 8x2 y

by 8X = -

+ = -

= y3)

8( 8y

by = 0

+

8y3 -

8X

+

= 8yi 8y 0

+ =

- 1) =

y

8y(y8 -

-

y = X 0 -

=

= -

- y

1

1

y y

X = 1

= =

1

- y =

- =

=

1

-

1 1

x

y =

-

= (a

24x (

*

fxx 8

= zhy

Ayy :

ray

=

Axy 1yx

8 =

= ( 8) sens

H(oj) : 64co

= -

= /)

H(y 1)

1) ( n run

20

=

-

, =

= ,

+

y) y2

f(x +

x xy

=

, 2x

y

fx 2x = -

+ y

=

Ay +

2y x 4X X

= + =

- x 0

3x =

-

=

- y 0

=

Axx 2 (ii)

=

fey L

= byx

Axy 1 =

= )

) 4 3

1 =

-

=

0)

H(0 =

,