Critical point (mathematics)

See also: Critical point (set theory)

The abcissae of the red circles are stationary points; the blue squares are inflection points. It's important to note that the stationary points are critical points, but the inflection points are not nor are their abcissae.

In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0.^[1][2] The value of the function at a critical point is a critical value of the function. These definitions admit generalizations to functions of several variables^[3], differentiable maps between R^m and Rⁿ, and differentiable maps between differentiable manifolds.

Definition for single variable functions

A critical point of a function of a single real variable, ƒ(x), is a value x₀ in the domain of ƒ where either the function is not differentiable or its derivative is 0, ƒ^′(x₀) = 0. Any value in the codomain of ƒ that is the image of a critical point under ƒ is a critical value of ƒ. These concepts may be visualized through the graph of ƒ: at a critical point, either the graph does not admit the tangent or the tangent is a vertical or horizontal line. In the last case, the derivative is zero and the point is called a stationary point of the function.

Optimization

Main article: Maxima and minima

By Fermat's theorem, local maxima and minima of a function can occur only at its critical points. However, not every stationary point is a maximum or a minimum of the function — it may also correspond to an inflection point of the graph, as for ƒ(x) = x³ at x = 0, or the graph may oscillate in the neighborhood of the point, as in the case of the function defined by the formulae ƒ(x) = x²sin(1/x) for x ≠ 0 and ƒ(0) = 0, at the point x = 0.

Examples

• The function ƒ(x) = x² + 2x + 3 is differentiable everywhere, with the derivative ƒ^′(x) = 2x + 2. This function has a unique critical point −1, because it is the unique number x₀ for which 2x₀ + 2 = 0. This point is a global minimum of ƒ. The corresponding critical value is ƒ(−1) = 2. The graph of ƒ is a concave up parabola, the critical point is the abscissa of the vertex, where the tangent line is horizontal, and the critical value is the ordinate of the vertex and may be represented by the intersection of this tangent line and the y-axis.

• The function f(x) = x^2/3 is defined for all x and differentiable for x ≠ 0, with the derivative ƒ^′(x) = 2x^−1/3/3. Since ƒ^′(x) ≠ 0 for x ≠ 0, the only critical point of ƒ is x = 0. The graph of the function ƒ has a cusp at this point with vertical tangent. The corresponding critical value is ƒ(0) = 0.

• The function ƒ(x) = x³ − 3x + 1 is differentiable everywhere, with the derivative ƒ^′(x) = 3x² − 3. It has two critical points, at x = −1 and x = 1. The corresponding critical values are ƒ(−1) = 3, which is a local maximum value, and ƒ(1) = −1, which is a local minimum value of ƒ. This function has no global maximum or minimum. Since ƒ(2) = 3, we see that a critical value may also be attained at a non-critical point. Geometrically, this means that a horizontal tangent line to the graph at one point (x = −1) may intersect the graph at an acute angle at another point (x = 2).

Several variables

In this section, functions are assumed to be smooth.

For a smooth function of several real variables, the condition of being a critical point is equivalent to all of its partial derivatives being zero; for a function on a manifold, it is equivalent to its differential being zero.

If the Hessian matrix at a critical point is nonsingular then the critical point is called nondegenerate, and the signs of the eigenvalues of the Hessian determine the local behavior of the function. In the case of a real function of a real variable, the Hessian is simply the second derivative, and nonsingularity is equivalent to being nonzero. A nondegenerate critical point of a single-variable real function is a maximum if the second derivative is negative, and a minimum if it is positive. For a function of n variables, the number of negative eigenvalues of a critical point is called its index, and a maximum occurs when all eigenvalues are negative (index n, the Hessian is negative definite) and a minimum occurs when all eigenvalues are positive (index zero, the Hessian is positive definite); in all other cases, the critical point can be a maximum, a minimum or a saddle point (index strictly between 0 and n, the Hessian is indefinite). Morse theory applies these ideas to determination of topology of manifolds, both of finite and of infinite dimension.

Gradient vector field

In the presence of a Riemannian metric or a symplectic form, to every smooth function is associated a vector field (the gradient or Hamiltonian vector field). These vector fields vanish exactly at the critical points of the original function, and thus the critical points are stationary, i.e. constant trajectories of the flow associated to the vector field.

Definition for maps

For a differentiable map f between R^m and Rⁿ, critical points are the points where the differential of f is a linear map of rank less than n; in particular, every point is critical if m < n.^[4] This definition immediately extends to maps between smooth manifolds. The image of a critical point under f is a called a critical value. A point in the complement of the set of critical values is called a regular value. Sard's theorem states that the set of critical values of a smooth map has measure zero.

See also

• Singular point of a curve
• Singularity theory

References

1. ^ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5. 
2. ^ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 0-547-16702-4. 
3. ^ Adams, A. Adams; Essex, Christopher (2009). Calculus: A Complete Course. Pearson Prentice Hall. p. 744. ISBN 9780321549280. 
4. ^ Carmo, Manfredo Perdigão do (1976). Differential geometry of curves and surfaces. Upper Saddle River, NJ: Prentice-Hall. ISBN 0132125897.