Stationary points

Section: Introduction to Stationary Points

  • Stationary points occur where the surface’s rate of change is zero in all directions in the x-y plane – where the derivative of the function is equal to zero.

  • They hold a significant importance in this part of Further Mathematics as an understanding of their behaviour facilitates the analysis of surfaces.

  • Stationary points can be classified into distinct categories such as local maxima, local minima, and saddle points.

Section: Identifying Stationary Points

  • Stationary points can be identified by taking the first partial derivatives with respect to both variables and equating them to zero.

  • By solving this system of equations simultaneously, the coordinates of stationary points on the surface can be determined.

Section: Classification of Stationary Points

  • The nature of a stationary point, whether it is a maximum, minimum or saddle point, can be determined through the second derivative test.

  • The second derivative test uses the second partial derivatives of the function and the determinant of the Hessian matrix.

  • If the determinant is positive and the second derivative with respect to x (and y) is positive, the point is a local minimum.

  • If the determinant is positive and the second derivative with respect to x (and y) is negative, the point is a local maximum.

  • If the determinant is negative, the point is a saddle point.

Section: Plotting and Analysing Stationary Points

  • Once the stationary points have been located and classified, they can be used to gain insight into the overall shape and behaviour of the surface.

  • Stationary points can be analytically represented on a contour plot to visualise the points of maxima, minima and saddle points.

  • Recognition of the type of stationary point can aid in the understanding of modelling real world scenarios, making the topic of stationary points an integral aspect of Further Mathematics.