Second Derivative Test
After establishing the critical points of a function, the second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. If the function f is twice-differentiable at a critical point x (i.e. a point where f '(x) = 0), then:
- If , then has a local maximum at .
- If , then has a local minimum at .
- If , the test is inconclusive.
In the last case, Taylor's Theorem may sometimes be used to determine the behavior of f near x using higher derivatives.
Proof of the second-derivative test
Suppose we have (the proof for is analogous). By assumption, . Then
Thus, for h sufficiently small we get
which means that if (intuitively, f is decreasing as it approaches from the left), and that if (intuitively, f is increasing as we go right from x). Now, by the first-derivative test, has a local minimum at .
A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point. It does not, however, provide information about inflection points. Specifically, a twice-differentiable function f is concave up if and concave down if . Note that if , then has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is an inflection point.
Higher-order derivative test
The higher-order derivative test or general derivative test is able to determine whether a function's critical points are maxima, minima, or points of inflection for a wider variety of functions than the second-order derivative test. As shown below, the second-derivative test is mathematically identical to the special case of n = 1 in the higher-order derivative test.
Let f be a real-valued, sufficiently differentiable function on an interval , let , and let be a natural number. Also let all the derivatives of f at c be zero up to and including the n-th derivative, but with the (n + 1)th derivative being non-zero:
There are four possibilities, the first two cases where c is an extremum, the second two where c is a (local) saddle point:
- If n is odd and , then c is a local maximum.
- If n is odd and , then c is a local minimum.
- If n is even and , then c is a strictly decreasing point of inflection.
- If n is even and , then c is a strictly increasing point of inflection.
Since n must be either odd or even, this analytical test classifies any stationary point of f, so long as a nonzero derivative shows up eventually.
Say, we want to perform the general derivative test on the function at the point . To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero.
As shown above, at the point , the function has all of its derivatives at 0 equal to 0, except for the 6th derivative, which is positive. Thus n = 5, and by the test, there is a local minimum at 0.
- Second Derivative Test, The Organic Chemistry Tutor
- Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). New York: McGraw-Hill. pp. 231–267. ISBN 0-07-010813-7.
- Marsden, Jerrold; Weinstein, Alan (1985). Calculus I (2nd ed.). New York: Springer. pp. 139–199. ISBN 0-387-90974-5.
- Shockley, James E. (1976). The Brief Calculus : with Applications in the Social Sciences (2nd ed.). New York: Holt, Rinehart & Winston. pp. 77–109. ISBN 0-03-089397-6.
- Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8.
- Willard, Stephen (1976). Calculus and its Applications. Boston: Prindle, Weber & Schmidt. pp. 103–145. ISBN 0-87150-203-8.