Second Derivative Test

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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 .

Concavity test

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.



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