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In my calculus III textbook, the following sentence is causing trouble for me and preventing me from understanding the theory behind Lagrange multipliers.

"Since the gradient vector for a given function is orthogonal to its level curves at any given point, for a level curve of $f$ to be tangent to the constraint curve $g(x,y) = 0$, the gradients of $f$ and $g$ must be parallel"

There are bits and pieces I understand, but I'm missing the holistic picture that will put my mind at ease. I'm quite certain that I understand that for a given curve $f(x,y)$, its gradient will be tangent to the level surface $f(x,y,z) = k$ because its directional derivative will be $0$. Specifically, I'm hung up on the idea that they must be parallel I cannot directly see how the case where they are anti-parallel isn't possible. Furthermore, I'm not sure why the constraint curve $g(x,y)$ is set to $0$ in this explanation. If someone could explain in detail the ideas behind this sentence, I would appreciate it.

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The constraint curve can be given by $g(x,y)=c,$ where $c$ is any constant. But if you consider $h(x,y)=g(x,y)-c$ then you have that the curve is given by $h(x,y)=0.$ So, you always can assume that $g(x,y)=0.$

Now, since $g(x,y)=0,$ assuming that in a neigbourhood of a point it is $y=y(x),$ we get $$\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}\frac{dy}{dx}=0.$$ Since $T=\left(1,\frac{dy}{dx}\right)$ is tangent to the curve it is $T\cdot \nabla g=0,$ that is, $\nabla g$ is perpendicular to the curve.

Now, if we consider a level curve of $f$ we have the same result, that is, $\nabla f$ is perpendicular to such a curve. So, if a level curve of is tangent to the curve $g(x,y)=0$ at some point $(x_0,y_0)$ then both curves have the same tangent vector $T$ at $(x_0,y_0).$ Since $\nabla f\perp T$ (since $\nabla f$ is perpendicular to the tangent vector of its level curves) and $\nabla g\perp T,$ as we have seen, the only possibility is $\nabla f||\nabla g.$

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I would regard "parallel" here being used more loosely than you're reading it; there's nothing wrong with $\nabla f$ being anti-parallel. All that's important is that the gradients are scalar multiples of each other. But, it wouldn't surprise me if some definition you're given ensures that these gradients point in the same (as opposed to opposite) directions.

A constraint can always be made to equal zero simply by adding or subtracting a constant from the constraint function. Since that constant in no way affects the gradient, this represents a freedom that we can eliminate by setting the value of the constraint to zero, or to any other convenient constant.

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