A practical optimum must be feasible and must beat all relevant stationary and endpoint candidates. A derivative tests interior variation. It does not test an included boundary, so a real design range may put its best value at an endpoint. Method Define variables and constraint; eliminate one variable; state domain; differentiate; check stationary candidates; compare included endpoints or excluded-end limits; return the requested quantity with units. Example A=x(20-2x), 0<x<10, A =20-4x gives x=5, A=50; A =-4<0 and both endpoint limits are zero. Exam check A zero derivative alone does not prove a maximum. An excluded endpoint can give a limiting value without an attained optimum.

A practical optimum must be feasible and must beat all relevant stationary and endpoint candidates.

A derivative tests interior variation. It does not test an included boundary, so a real design range may put its best value at an endpoint.

Define variables and constraint; eliminate one variable; state domain; differentiate; check stationary candidates; compare included endpoints or excluded-end limits; return the requested quantity with units.

\(A=x(20-2x),\quad0<x<10,\quad A^{\prime}=20-4x\) gives \(x=5\), \(A=50\); \(A^{\prime\prime}=-4<0\) and both endpoint limits are zero.

A zero derivative alone does not prove a maximum. An excluded endpoint can give a limiting value without an attained optimum.

Practical optimisation and endpoints

Practical optimisation and endpoints