Second derivative test

In calculus, the second derivative test is a criterion often useful for determining whether a given stationary point of a function is a local maximum or a local minimum using the value of the second derivative at the point.

The test states: If the function f is twice differentiable at a stationary point x, meaning that $\ f^{\prime}(x) = 0$, then:

• If $\ f^{\prime\prime}(x) < 0$ then $\ f$ has a local maximum at $\ x$.
• If $\ f^{\prime\prime}(x) > 0$ then $\ f$ has a local minimum at $\ x$.
• If $\ f^{\prime\prime}(x) = 0$, the second derivative test says nothing about the point $\ x$, a possible inflection point.

In the last case, although the function may have a local maximum or minimum at x, because the function is sufficiently "flat" (i.e. $\ f^{\prime\prime}(x) = 0$) the extremum is rendered undetected by the second derivative. In this case one has to examine the third derivative. The point at which $\ f^{\prime\prime}(x) = 0$ is an inflection point if concavity changes on either side of it. For example, (0,0) is an inflection point on $\ f(x) = x^3$ because $\ f^{\prime\prime}(0) = 0$, and $\ f^{\prime\prime}(-1) < 0$ and $\ f^{\prime\prime}(1) > 0$.

Multivariable case

Main article: Second partial derivative test

For a function of more than one variable, the second derivative test generalizes to a test based on the eigenvalues of the function's Hessian matrix at the stationary point. In particular, assuming that all second order partial derivatives of f are continuous on a neighbourhood of a stationary point x, then if the eigenvalues of the Hessian at x are all positive, then x is a local minimum. If the eigenvalues are all negative, then x is a local maximum, and if some are positive and some negative, then the point is a saddle point. If the Hessian matrix is singular, then the second derivative test is inconclusive.

Proof of the second derivative test

Suppose we have f''(x) > 0 (the proof for f''(x) < 0 is analogous). Then

$0 < f''(x) = \lim_{h \to 0} \frac{f'(x + h) - f'(x)}{h} = \lim_{h \to 0} \frac{f'(x + h) - 0}{h} = \lim_{h \to 0} \frac{f'(x+h)}{h}.$

Thus, for h sufficiently small we get

$\frac{f'(x+h)}{h} > 0$

which means that f'(x + h) < 0 if h < 0 so that f is decreasing to the left of x, and that f'(x + h) > 0 if h > 0 so that f is increasing to the right of x.

Now, by the first derivative test we know that f has a local minimum at x.

Concavity test

The second derivative test may also be used to determine the concavity of a function as well as a function's points of inflection.

First, all points at which $\ f''(x) = 0$ are found. In each of the intervals created, $\ f''(x)$ is then evaluated at a single point. For the intervals where the evaluated value of $\ f''(x) < 0$ the function $\ f(x)$ is concave down, and for all intervals between critical points where the evaluated value of $\ f''(x) > 0$ the function $\ f(x)$ is concave up. The points that separate intervals of opposing concavity are points of inflection.

See also

• Bordered Hessian
• First derivative test
• Optimization (mathematics)
• Fermat's theorem
• Higher-order derivative test
• Differentiability
• Extreme value
• Inflection point
• Convex function
• Concave function

References

• Second Derivative Test from Mathworld
• Concavity and the Second Derivative Test
• Thomas Simpson's use of Second Derivative Test to Find Maxima and Minima at Convergence