Calculus BC: Applications of the Derivative

Analysis of Graphs

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 (x ₀, f (x ₀)) on the graph of f such that f (x ₀)≤f (x) (resp. f (x ₀)≥f (x) ) for all x in some interval containing x ₀ . 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).

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) = x ³ has a critical point at 0 which is of this type. The derivative f'(x) = 3x ² is zero here, but everywhere else f' is positive. This function and its derivative are sketched below.

The Second Derivative Test

Once we have found the critical points, one way to determine if they are local minima or maxima is to apply the first derivative test. Another way uses the second derivative of f . Suppose x ₀ is a critical point of the function f (x) , that is, f'(x ₀) = 0 . We have the following three cases:

1. f''(x ₀) > 0 implies x ₀ is a local minimum
2. f''(x ₀) < 0 implies x ₀ is a local maximum
3. f''(x ₀) = 0 is inconclusive

The first two of these options derive from the observation that f''(x ₀) is the rate of change of f'(x) at x ₀ , which will be positive if the derivative crosses zero from negative to positive, and negative is the derivative crosses zero from positive to negative. This is called the second derivative test for maxima and minima. The third, inconclusive case is considered below.

The first and second derivative tests employ essentially the same logic, examining what happens to the derivative f'(x) near a critical point x ₀ . The first derivative test says that maxima and minima correspond to f' crossing zero from one direction or the other, which is indicated by the sign of f' near x ₀ . The second derivative test is just the observation that the same information is encoded in the slope of the tangent line to f'(x) at x ₀ .

Concavity and Inflection Points

A function f (x) is called concave up at x ₀ if f''(x ₀) > 0 , and concave down if f''(x ₀) < 0 . Graphically, this represents which way the graph of f is "turning" near x ₀ . A function that is concave up at x ₀ lies above its tangent line in a small interval around x ₀ (touching but not crossing at x ₀ ). Similarly, a function that is concave down at x ₀ lies below its tangent line near x ₀ .

The remaining case is a point x ₀ where f''(x ₀) = 0 , which is called an inflection point. At such a point the function f holds closer to its tangent line than elsewhere, since the second derivative represents the rate at which the function turns away from the tangent line. Put another way, a function usually has the same value and derivative as its tangent line at the point of tangency; at an inflection point, the second derivatives of the function and its tangent line also agree. Of course, the second derivative of the tangent line function is always zero, so this statement is just that f''(x ₀) = 0 .

Inflection points are the critical points of the first derivative f'(x) . At an inflection point, a function may change from being concave up to concave down (or the other way around), or momentarily "straighten out" while having the same concavity to either side. These three cases correspond, respectively, to the inflection point x ₀ being a local maximum or local minimum of f'(x) , or neither.