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My textbook states that when finding the maxes or mins of some function f(x,y,z) with some constraining equation g(x,y,z) that g must be made to equal zero. I would like to know why when it is the gradient of g that we care about. Why does it matter? Thanks.

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  • $\begingroup$ Presumably the constraint is $g(x,y,z) = 0$, so it must hold. $\endgroup$
    – copper.hat
    Commented Nov 4, 2015 at 6:41
  • $\begingroup$ I don't believe that the constraining equation must equal zero. Perhaps the textbook was assuming an example in which the constraint $g(x,y,z)$ was zero? $\endgroup$
    – Simon
    Commented Nov 4, 2015 at 7:30
  • $\begingroup$ the textbook would always rearrange the constraint equation so that they could then define g(x,y,z)=0, and then go on to talk about how the gradient of g(x,y,z) = gradient of f(x,y,z) when they both = 0 (at maxes or mins). I get that. I don't know what the big deal is about setting g(x,y,z)=0. For example say the constraint is y = 15z - x^58 +5. They insist on writing and assigning g(x,y,z)= y + x^58 -15z -5 =0.Are there any ramifications of leaving it as g(x,y,z) =y+ x^58 -15z = 5? $\endgroup$
    – user11585
    Commented Nov 4, 2015 at 7:46

1 Answer 1

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Consider the problem to minimize $f(x) = x^2$ such that $x=3$. There is only one solution, namely $x=3$. But $\nabla f(x) \neq 0$ there.

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