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Normally lagrange multipliers are used in the following sense.

Suppose we are given a function f(x,y.z..,) and the constraint g(x,y,z,...,) = c

Define a lagrange function:

L = f - λ(g-c)

And find the partial derivatives with respect to all variables and λ. This gives you the extrema since for an extrema ∇f = λ∇g

However, I find that in my mechanics book this is used differently. I have attached a picture of the place where lagrange multipliers are used. I don't see how they in this case are used to maximize a quantity (which should be L) under a constraint like the above. Can anyone show me how they are and which gradients are to be parallel?

Suppose we are given a function f(x,y.z..,) and the constraint g(x,y,z,...,) = c

Define a lagrange function:

L = f - λ(g-c)

And find the partial derivatives with respect to all variables and λ. This gives you the extrema since for an extrema ∇f = λ∇g

However, I find that in my mechanics book this is used differently. I have attached a picture of the place where lagrange multipliers are used. I don't see how they in this case are used to maximize a quantity (which should be L) under a constraint like the above. Can anyone show me how they are and which gradients are to be parallel?