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I was studying Lagrange multipliers. However, I have some confusion. Let's say I have a function $f(x,y)$ to be minimized and I have some constraints $g(x,y) = 0$.

If I minimize the function $$ L(x,y,\lambda) = f(x,y) + \lambda g(x,y) \>, $$ then how does it include the constraint $g(x,y) = 0$. The book says that if I minimize $L$ with respect to $\lambda$ then it will be equivalent to minimize the function $f(x,y)$ with the constraint $g(x,y)$.

I need some clarifications.

Further it is said that

gradient(f)+ lambda * gradient(g) = 0 ............(1) 

leads to

L(x,y,lambda) = f(x,y) + lambda * g(x,y)...........(2) 

I didn't get this portion how come equation 1 led to equation 2?

Also I am a bit confused when it comes to inequality constraints like

g(x,y) >= 0 

It is being said that f(x,y) will be maximum if its gradient is oriented away from the region g(x,y) > 0 and therefore

gradient(f(x,y)) = - lambda * gradient(g(x,y)) 

I just didn't get this.

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    Please remark that the method of Lagrange multipliers simply gives a condition to find **critical points** of $f$ constrained to $g^{-1}(0)$. Free critical points of $L$ needn't be minima.2012-06-24

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