Calculus BC: Applications of the Derivative: Analysis of Graphs

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Derivatives can be used to gather information about the graph of a function. Since the derivative represents the rate of change of a function, to determine when a function is increasing, we simply check where its derivative is positive. Similarly, to find when a function is decreasing, we check where its derivative is negative.

The points where the derivative is equal to 0 are called critical points. At these points, the function is instantaneously constant and its graph has horizontal tangent line. For a function representing the motion of an object, these are the points where the object is momentarily at rest.

The First Derivative Test

A local minimum (resp. local maximum) of a function f is a point (x₀, f (x₀)) on the graph of f such that f (x₀)≤f (x) (resp. f (x₀)≥f (x)) for all x in some interval containing x₀. Such a point is called a global minimum (resp. global maximum) of a function f if the appropriate inequality holds for all points in the domain. In particular, any global maximum (minimum) is also a local maximum (minimum).

It is intuitively clear that the tangent line to the graph of a function at a local minimum or maximum must be horizontal, so the derivative at the point is 0, and the point is a critical point. Therefore, in order to find the local minima/maxima of a function, we simply have to find all its critical points and then check each one to see whether it is a local minimum, a local maximum, or neither. If the function has a global minimum or maximum, it will be the least (resp. greatest) of the local minima (resp. maxima), or the value of the function on an endpoint of its domain (if any such points exist).

Figure %: Examples of Global and Local Extrema

Clearly, the behavior near a local maximum is that the function increases, levels off, and begins decreasing. Therefore, a critical point is a local maximum if the derivative is positive just to the left of it, and negative just to the right. Similarly, a critical point is a local minimum if the derivative is negative just to the left and positive to the right. These criteria are collectively called the first derivative test for maxima and minima.

There may be critical points of a function that are neither local maxima or minima, where the derivative attains the value zero without crossing from positive to negative. For instance, the function f (x) = x³ has a critical point at 0 which is of this type. The derivative f'(x) = 3x² is zero here, but everywhere else f' is positive. This function and its derivative are sketched below.