Principles of Optimization Using Derivatives

The second derivative test can help confirm whether a critical point is a local maximum or minimum.

The first derivative test involves checking the sign changes of $f'(x)$ around critical points.

An optimization problem can be solved solely by setting the first derivative equal to zero, with no need to consider endpoints or verify the nature of the critical points.

For optimization on a closed interval, the optimal value occurs either at a critical point or at an endpoint.