Which of the following can NOT be determined by differentiating the functional form of a curve?

The determination of the area under a curve between certain bounds typically involves integration rather than differentiation. While differentiation is used to find the slope or the rate of change of a function at a given point, calculating the area under a curve requires accumulating these changes over an interval, which is the fundamental aspect of integration.

In contrast, the concavity of a curve can indeed be found by analyzing the second derivative of the function—if the second derivative is positive, the curve is concave up, and if it's negative, concave down. Similarly, the location of inflection points, where the curve changes from concave up to concave down (or vice versa), is found by identifying where the second derivative equals zero, indicating a potential change in concavity. The number of inflection points can also be assessed through differentiating, as it relies on finding where the second derivative is zero or undefined and analyzing its sign changes.

Therefore, the only option that cannot be determined through differentiation processes is the area under the curve, as this is fundamentally addressed through the principles of integration.