Critical point (mathematics)

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In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below). The value of the function at a critical point is a critical value.^[1]

More specifically, when dealing with functions of a real variable, a critical point, also known as a stationary point, is a point in the domain of the function where the function derivative is equal to zero (or where the function is not differentiable).^[2] Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not holomorphic).^[3][4] Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined).^[5]

This sort of definition extends to differentiable maps between ⁠${\displaystyle \mathbb {R} ^{m}}$⁠ and ⁠${\displaystyle \mathbb {R} ^{n},}$⁠ a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable maps between differentiable manifolds, as the points where the rank of the Jacobian matrix decreases. In this case, critical points are also called bifurcation points. In particular, if C is a plane curve, defined by an implicit equation f (x,y) = 0, the critical points of the projection onto the x-axis, parallel to the y-axis are the points where the tangent to C are parallel to the y-axis, that is the points where ${\textstyle {\frac {\partial f}{\partial y}}(x,y)=0}$. In other words, the critical points are those where the implicit function theorem does not apply.