Understanding Stationary Points

A stationary point is a point on a graph where the derivative or gradient is zero.
There are three types of stationary points: maximum points, minimum points, and points of inflection.
A function has a maximum at a point if it changes from increasing to decreasing at that point.
A function has a minimum at a point if it changes from decreasing to increasing at that point.
In a point of inflection, the function changes the nature of its curvature.
To find stationary points, set the derivative equal to zero and solve for x.

Classifying Stationary Points

The second derivative can be used to classify the nature of a stationary point.
If the second derivative is greater than zero at a point, that point is a local minimum.
If the second derivative is less than zero at a point, that point is a local maximum.
If the second derivative is equal to zero at a point, the test is inconclusive, and you need to refer to the gradient of points close by.
This process is known as the Second Derivative Test.

Understanding stationary points is crucial for solving practical problems in various fields.
Stationary points can help determine optimum values, like maximum profit or minimum cost in economics.
In physics and engineering, they help in understanding the points of equilibrium.
They can give key insights into polynomial and trigonometric functions in mathematics.

First order derivative: If y = f(x), then the first derivative is given by dy/dx = f’(x).
Second order derivative: If y’ = f’(x), then the second derivative is given by d²y/dx² = f’‘(x).
Setting derivative equal to zero for stationary points: f’(x) = 0.