To ascertain whether a point is a maximum or minimum, we can employ the first and second derivative tests.

The first derivative test involves calculating the derivative of the function and setting it equal to zero to identify critical points. These critical points occur where the slope of the function is zero, suggesting the presence of potential maxima or minima. Once the critical points are determined, we analyze the sign of the derivative both before and after each point. If the derivative transitions from positive to negative, the point is classified as a local maximum. Conversely, if the derivative shifts from negative to positive, the point is identified as a local minimum.

The second derivative test offers an alternative approach. Following the identification of critical points using the first derivative, we then compute the second derivative of the function. If the second derivative at a critical point is positive, the function is concave up at that point, indicating a local minimum. If the second derivative is negative, the function is concave down, suggesting a local maximum. In cases where the second derivative equals zero, the test becomes inconclusive, necessitating the use of additional methods to determine the nature of the critical point.

These techniques facilitate a deeper understanding of the function’s behavior around critical points, thereby simplifying the process of identifying whether they correspond to maxima or minima.