多元函数极值与 A A A及 Δ = B 2 − A C \Delta =B^2-AC Δ=B2−AC的关系
结论:
Δ = B 2 − A C > 0 \Delta =B^2-AC > 0 Δ=B2−AC>0时, A > 0 A>0 A>0取极小值, A < 0 A<0 A<0取极大值
Δ = B 2 − A C < 0 \Delta =B^2-AC < 0 Δ=B2−AC<0时,无极值
Δ = B 2 − A C = 0 \Delta =B^2-AC = 0 Δ=B2−AC=0时,需要进一步讨论,一般从极值定义去讨论
那么,这个结论是怎么来的?
对于一元函数 f ′ ( x ) = 0 , f ′ ′ ( x ) > 0 f'(x) = 0,f''(x)>0 f′(x)=0,f′′(x)>0取极小值, f ′ ′ ( x ) < 0 f''(x)<0 f′′(x)<0取极大值。
对于多元函数 ∂ f ∂ l = 0 , ∂ 2 f ∂ l 2 > 0 \dfrac{\partial f}{\partial l} = 0,\dfrac{\partial^2 f}{\partial l^2}>0 ∂l∂f=0,∂l2∂2f>0取极小值, ∂ 2 f ∂ l 2 < 0 \dfrac{\partial^2 f}{\partial l^2}<0 ∂l2∂2f<0取极大值。
求多元函数极值时,我们关注的不是偏导数而是方向导数,因为 f ( x , y ) f(x,y) f(x,y)沿任意方向都可以变化,而偏导数只描述了沿x,y方向的变化,方向导数则可描述随意方向。
∂ f ∂ l = ▽ f ⋅ ( c o s α , c o s β ) = f x ′ c o s α + f y ′ c o s β \dfrac{\partial f}{\partial l} = \bigtriangledown f·(cos\alpha,cos\beta) = f'_xcos\alpha+f'_ycos\beta ∂l∂f=▽f⋅(cosα,cosβ)=fx′cosα+fy′cosβ
令 g ( x , y ) = ∂ f ∂ l = f x ′ c o s α + f y ′ c o s β g(x,y) = \dfrac{\partial f}{\partial l} = f'_xcos\alpha+f'_ycos\beta g(x,y)=∂l∂f=fx′cosα+fy′cosβ
∂ g ∂ l = ▽ g ⋅ ( c o s α , c o s β ) = c o s 2 β [ f x x ′ ′ ( c o s α c o s β ) 2 + 2 f x y ′ ′ c o s α c o s β + f y y ′ ′ ] \dfrac{\partial g}{\partial l} = \bigtriangledown g·(cos\alpha,cos\beta) = cos^2\beta[f''_{xx}(\dfrac{cos\alpha}{cos\beta})^2+2f''_{xy}\dfrac{cos\alpha}{cos\beta}+f''_{yy}] ∂l∂g=▽g⋅(cosα,cosβ)=cos2β[fxx′′(cosβcosα)2+2fxy′′cosβcosα+fyy′′]
∂ 2 f ∂ l 2 = c o s 2 β [ f x x ′ ′ ( c o s α c o s β ) 2 + 2 f x y ′ ′ c o s α c o s β + f y y ′ ′ ] \dfrac{\partial^2 f}{\partial l^2} =cos^2\beta[f''_{xx}(\dfrac{cos\alpha}{cos\beta})^2+2f''_{xy}\dfrac{cos\alpha}{cos\beta}+f''_{yy}] ∂l2∂2f=cos2β[fxx′′(cosβcosα)2+2fxy′′cosβcosα+fyy′′]
令 A = f x x ′ ′ , B = f x y ′ ′ , C = f y y ′ ′ A =f''_{xx},B =f''_{xy},C = f''_{yy} A=fxx′′,B=fxy′′,C=fyy′′
要使得 ∂ 2 f ∂ l 2 > 0 \dfrac{\partial^2 f}{\partial l^2}>0 ∂l2∂2f>0,则 A > 0 , Δ < 0 A>0,\Delta<0 A>0,Δ<0,应该注意的是 Δ < 0 \Delta<0 Δ<0这个条件是必须满足的,因为 ∂ 2 f ∂ l 2 > 0 \dfrac{\partial^2 f}{\partial l^2}>0 ∂l2∂2f>0是一个恒成立问题。
其他情况同理。