Optimisation with a Restricted Domain

For an absolute maximum or minimum on a closed interval, compare all admissible stationary values and both endpoints. A stationary point outside the physical domain cannot be used.

Example

For \(V(x)=x(10-x)\) with \(0\le x\le3\), \(V^{\prime}(x)=10-2x\). The stationary value \(x=5\) is outside the domain.

\(V(0)=0,\qquad V(3)=21\). Since \(V^{\prime}(x)>0\) throughout the interior, the maximum is \(21\) at the endpoint \(x=3\).

When the constraint defines variables implicitly, differentiate that constraint before substituting the instantaneous condition. Use the chain rule to connect the derivative of the objective to the independent variable.