Critical Point (mathematics) - Examples

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) = x2/3 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).
• The function ƒ(x) = 1/x has one critical point. The point x = 0 is a critical point because it is an isolated point between the domain.