Applications of Derivatives

Finding the Best

One of the most useful things derivatives do is optimisation — finding where a quantity is largest or smallest. Because the tangent is horizontal at a peak or a valley, the derivative is zero at these interior extreme points. This turns "find the maximum" into "solve $f'(x) = 0$."

The systematic method:

1. Find the critical points — where $f'(x) = 0$ (or where the derivative does not exist).
2. Classify each: a maximum, a minimum, or neither.
3. On a closed interval, also check the endpoints, since the extreme value can occur there even when the derivative is not zero.

To classify a critical point, use the derivative's sign or the second derivative:

• First-derivative test — if $f'$ changes from $+$ to $-$ at the point, it is a local maximum (rising then falling); from $-$ to $+$, a local minimum.
• Second-derivative test — the second derivative $f''$ measures concavity. $f'' > 0$ means the curve is concave up (cup-shaped, so a critical point there is a minimum); $f'' < 0$ means concave down (a maximum). Where $f''$ changes sign is an inflection point.

This is the mathematical core of a vast range of computing: training a machine-learning model minimises a loss function, resource allocation maximises utility, and countless engineering designs optimise some objective. Gradient descent (next lessons' multivariable version) is exactly "follow the derivative downhill to a minimum."

Common pitfall: assuming every point with $f'(x) = 0$ is a maximum or minimum, and forgetting the endpoints. A zero derivative marks a critical point, but it could be an inflection point (like $x^3$ at 0, which is neither a max nor a min) — you must classify it. And on a closed interval the true extreme may sit at an endpoint, where the derivative need not be zero, so checking only $f'(x)=0$ can miss the actual maximum.