Critical Points

Critical points refer to locations where the derivative of a function is zero or undefined. Among critical points, there are two important types: stationary points and singular points.

Stationary Points

These points occur when the derivative of a function is zero. In other words, the rate of change of the function at these points is horizontal (flat) because the slope of the curve is zero. Stationary points include maxima, minima, and points of inflection.

• Maxima: Points where the function reaches a local maximum value.
• Minima: Points where the function reaches a local minimum value.
• Inflection Points (“Saddle points”): Points where the function is neither a maximum nor a minimum but rather a point of inflection.

Singular Points

These points happen when the derivative of a function is undefined. They can occur due to various reasons, such as a vertical tangent or a cusp in the curve. Singular points are essential in understanding the behavior of functions, especially when dealing with functions that possess irregular or undefined behavior.

Example

Let's consider the function f(x)= x^3 − 3x.

To find the stationary points, we’ll first take the derivative of the function: $f\prime (x)=3x^2-3$

Next, we’ll set the derivative equal to zero to find the Critical Points: $3x^2-3=0$

Solving for $x$: $x^2-1=0$ $x^2=1$ $x=\pm 1$