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Critical point (mathematics)


In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0. Some authors include in the critical points the limit points where the function may be prolongated by continuity and where the derivative is not defined. For a differentiable function of several real variables, a critical point is a value in its domain where all partial derivatives are zero. The value of the function at a critical point is a critical value.

The interest of this notion lies in the fact that the points where the function has local extrema are critical points.

This definition extends to differentiable maps between Rm and Rn, 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 In other words, the critical points are those where the implicit function theorem does not apply.


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