# 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.

## Contents

## 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.

## Example

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.

## Resources

- Second Derivative Test, The Organic Chemistry Tutor

## References

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- 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.