Optimization

To maximise or minimise a quantity that depends on a variable, turn the problem into a function and use calculus. First write the quantity (area, volume, cost, etc.) as a function $f(x)$. Then find its derivative $f^{\prime }(x)$ and locate the points where $f^{\prime }(x)=0$ – these are the *critical points*. Finally, decide whether each critical point gives a maximum, a minimum, or neither by checking the sign of $f^{\prime }(x)$ on either side (the first‑derivative test) or by using the second derivative $f^{\prime \prime }(x)$. The largest (or smallest) value of $f$ among the admissible critical points and any boundary values is the optimum.