082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

846 Chapter 25 Functions of several variables
EXERCISES25.7
1 Determine the position and nature ofthe stationary
pointsofthe following functions:
(a) f(x,y)=x+y+xy
(b) f(x,y)=x 2 +y 2 −2y
(c) f(x,y)=x 2 +xy−y
(d) f(x,y)=x 2 +y 2 −xy
(e) f(x,y)=x 2 +2y 2 −3xy+x
2 Determine the position and nature ofthe stationary
pointsofthe following functions:
(a) z(x,y) =x 2 +y 2 −3xy +2x
(b) z(x,y) =x 3 +xy +y 2
(c) z(x,y) = y3 3 −x2 −y
(d) z(x,y) = 1 x + 1 y − 1 xy
(e) z(x,y) = 4x 2 y −6xy
3 Locateandidentify the stationary pointsofthe
following:
(a) f(x,y)=x 2 y−x 2 +y 2
(b) f(x,y) = x y +x+y
(c) f(x,y) =x 4 +16xy +y 4
(d) f(x,y)=y−y 2 −e x x
(e) f(x,y) = ex
y 4 −xy
Solutions
1 (a) (−1, −1),saddle point
(b) (0,1),minimum
(c) (1, −2),saddle point
(d) (0,0),minimum
(e) (4,3),saddle point
(
2 (a) 45
, 5)
6 ,saddle point
)
,minimum
(
(b) (0,0),saddle point; 16
, −12
1
(c) (0,1),saddle point; (0, −1),maximum
(d) (1,1),saddle point
(e) (0,0), (1.5,0),saddlepoints
3 (a) (0,0),saddle point
(b) (1,−1),saddle point
(c) (0,0),saddle point; (2,−2),minimum; (−2,2),
minimum
(d) (−1,0.5),maximum
(e) (−4,0.4493), saddlepoint
REVIEWEXERCISES25
1 Find all firstpartialderivatives ofthe following
functions:
(a) z(x,y) = 9x 2 +2y 2
(b) z(x,y) = 3x 3 y 6
(c) z(x,y) = 4x 3 −5y 3
(d) z(x,y) =xy +x 2 y 2
(e) z(x,y) = sin(2xy)
(f) z(x,y) = 2e 3xy
2 Evaluate the firstpartialderivatives of f atx = −1,
y=2.
(a) f(x,y) =3x 3 −y 3 +x 2 y 2
(b) f(x,y) = 4x
y
(c) f(x,y)=sinx+2cosy
(d) f(x,y) = (xy) 3 +x
(e) f(x,y) = x +y
x −y
(f) f(x,y)=2e x e y
3 Findallsecond partialderivatives ofzwhere
(a) z(x,y) = 3x 4 −9y 4 +x 3 y 3
(b) z(x,y) = (x 2 +y 2 ) 2
(c) z(x,y) = 2e 4x−3y
(d) z(x,y) = 2
x +y
(e) z(x,y) = 2 √ x +y
(f) z(x,y) = (sinx)(cosy)