Partial differentiation: Stationary points

Partial differentiation: Stationary points

Understanding Stationary Points

  • Stationary points are points in a graph of a function where the derivative equals zero.
  • If we consider a function z = f(x,y), a stationary point occurs for this function when both the first-order partial derivatives (∂f/∂x and ∂f/∂y) equal zero.
  • The method of finding stationary points for a function of one variable (i.e., f’(x) = 0) extends to functions of two variables.
  • Stationary points can be of three types: local maximum, local minimum or saddle points.

Partial Derivatives

  • Partial differentiation is the differentiation of multivariable functions.
  • If a function has more than one independent variable, we can find how the function changes along each of the axes by taking the derivative with respect to each of the variables, one at a time. These are called partial derivatives.
  • The symbol for a partial derivative is ∂. So, ∂f/∂x means partial derivative of f with respect to x.

Techniques for Finding Stationary Points

  • To find the stationary points of a function f(x,y), we first find the partial derivatives ∂f/∂x and ∂f/∂y.
  • We then set both of these derivatives equal to zero and solve the resulting equations to find the values of (x, y). This gives us the coordinate(s) of the stationary points.

Classification of Stationary Points

  • At a stationary point, we can determine the nature of the point (whether it’s a maximum, minimum, or neither) by using the second derivative test.
  • Compute the second order partial derivatives ∂²f/∂x², ∂²f/∂y² and ∂²f/∂x∂y.
  • Calculate D which is equal to ( ∂²f/∂x²)(∂²f/∂y²) - ( ∂²f/∂x∂y)².
  • If D > 0 and ∂²f/∂x² > 0, this indicates a local minimum. If D > 0 and ∂²f/∂x² < 0, this indicates a local maximum. If D < 0, it is a saddle point. If D = 0, the test is inconclusive.

Applications of Stationary Points

  • Stationary points can be used to find optimal solutions in mathematical problems,
  • They are useful for understanding the behaviour of physical and mathematical systems.
  • Stationary points are fundamental in optimisation problems, and feature extensively in fields such as economics, physics, and engineering.