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

848 Chapter 25 Functions of several variables
4
4 4 4
∂f
(d)
(x +y) 3, (x +y) 3, (x +y) 3
5
∂x = (2ax +cy)n(ax2 +by 2 +cxy) n−1 ,
(e) − 1 2 (x +y)−3/2 ,
∂f
−2 1 (x ∂y = (2by +cx)n(ax2 +by 2 +cxy) n−1
+y)−3/2 ,
−2 1 11 (a)27x
(x 2 −36y 2 −72xy +108x +276y −432
+y)−3/2
(b)p
(f) −(sinx)(cosy),
2 (2.1,2.9) = −26.97,
f (2.1,2.9) = −26.985
−(cosx)(siny),
12 x 2 −xy−2x+2y+2
−(sinx)(cosy)
∂f
∂x =an(ax +by)n−1 ,
13 y2
2 +x
∂f
14 (a) (0,1)minimum; (0, −1)saddle point
∂y =bn(ax +by)n−1 ,
(b) (0,0),(4,0)saddlepoints
∂ 2 f
(c) (2,0)minimum; (−2,0)saddle point
∂x 2 =a2 n(n −1)(ax +by) n−2 ,
( )
(d) (0,0)saddlepoint; 16
, 12
1 maximum
∂ 2 f
( ) ( )
∂x∂y =abn(n −1)(ax +by)n−2 ,
1
15 √ , − 1
2
∂ 2 2 √ , − 1 1
√ ,
2 2 2 √ ;saddlepoints
2
f
∂y 2 =b2 n(n −1)(ax +by) n−2