采用拉格朗日乘法证明：当(x=y=frac{c}{2})时，(xy)取得最大值
已知，C是任意大于零的常数，且 x + y = c ，证明当 (x=y=frac{c}{2}) 时，(xy) 取得最大值。
[ left{
egin{align}
& f(x,y) = xy \
& phi(x,y) = x + y -c = 0
end{align}
ight.
]
先作拉格朗日函数
[egin{align}
L(x,y) = f(x,y)+ lambdaphi(x,y) \
end{align}
]
求得
[ left{
egin{align}
& f_{x}(x,y) + lambdaphi_{x}(x,y) = 0 \
& f_{y}(x,y) + lambdaphi_{y}(x,y) = 0 \
& phi(x,y) = x + y -c = 0
end{align}
ight.
]
即
[ left{
egin{align}
& y + lambda = 0 \
& x + lambda = 0 \
& x + y = c
end{align}
ight.
]
解得
[ left{
egin{align}
& x = frac{c}{2} \
& y = frac{c}{2}
end{align}
ight.
]
证毕