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Stationary Points 18 - Loughborough University

Stationary Points calculation of the optimum value of a function of two variables is a common requirement in manyareas of engineering, for example in thermodynamics. Unlike the case of a function of one variablewe have to use more complicated criteria to distinguish between the various types of Stationary point. PrerequisitesBefore starting this Section you understand the idea of a function of twovariables be able to work out partial derivatives'&$%Learning OutcomesOn completion you should be able identify local maximum Points , localminimum Points and saddle Points on thesurfacez=f(x,y) use first partial derivatives to locate thestationary Points of a functionf(x,y) use second partial derivatives to determinethe nature of a Stationary pointHELM (2008):Section : Stationary Points211. The Stationary Points of a function of two variablesFigure 7 shows a computer generated picture of the surface defined by the functionz=x3+y3 3x 3y,where bothxandytake values in the interval[ , ].

The stationary value is f(0,2) = 0+24−16+5 = 13 Example 9 Find a second stationary point of f(x,y) = 8x2 +6y2 −2y3 +5. Solution f x = 16x and f y ≡ 6y(2 − y). From this we note that f x = 0 when x = 0, and f x = 0 and when y = 0, so x = 0, y = 0 i.e. (0,0) is a second stationary point of the function. It is important when solving the ...

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