why does the constraint in a

Question

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.

Answer 1

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.