Stationary Point - Curve Sketching

Curve Sketching

Determining the position and nature of stationary points aids in curve sketching, especially for continuous functions. Solving the equation f'(x) = 0 returns the x-coordinates of all stationary points; the y-coordinates are trivially the function values at those x-coordinates. The specific nature of a stationary point at x can in some cases be determined by examining the second derivative f''(x):

• If f''(x) < 0, the stationary point at x is concave down; a maximal extremum.
• If f''(x) > 0, the stationary point at x is concave up; a minimal extremum.
• If f''(x) = 0, the nature of the stationary point must be determined by way of other means, often by noting a sign change around that point provided the function values exist around that point.

A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points. However, this is limited again in that it works only for functions that are continuous in at least a small interval surrounding the stationary point.

A simple example of a point of inflection is the function f(x) = x3. There is a clear change of concavity about the point x = 0, and we can prove this by means of calculus. The second derivative of f is the everywhere-continuous 6x, and at x = 0, f′′ = 0, and the sign changes about this point. So x = 0 is a point of inflection.

More generally, the stationary points of a real valued function f: Rn → R are those points x₀ where the derivative in every direction equals zero, or equivalently, the gradient is zero.