SL 5.8—Extrema, optimization and inflexion

Optimization compares critical points with the feasible boundary.

A maximum or minimum occurs at a critical point or endpoint of the allowed domain; the model and constraints decide which is meaningful.

Example

For a rectangle with perimeter 20, A=x(10−x) is largest at x=5, not at an unconstrained value outside 0≤x≤10.

Differentiate, solve candidates, then compare all endpoints and check units.

A local maximum need not be the global maximum when the domain is restricted.

Classification: if $f'$ changes from positive to negative, the stationary point is a local maximum; negative to positive gives a local minimum. Alternatively, at $f'(a)=0$, $f''(a)>0$ implies a local minimum and $f''(a)<0$ a local maximum. A point of inflexion requires a change in concavity, so $f''=0$ alone is not sufficient; for example $y=x^4$ has $f''(0)=0$ but no concavity change. Use 'concave-up' for $f''>0$ and 'concave-down' for $f''<0$.