# Second derivative test

In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable 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 critical point x (i.e. f'(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 test is inconclusive.

In the latter case, Taylor's Theorem may be used to determine the behavior of f near x using higher derivatives.

## Multivariable case

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 critical point. In particular, assuming that all second order partial derivatives of f are continuous on a neighbourhood of a critical 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). By assumption, $f'(x) = 0$. 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 (intuitively, f is decreasing as it approaches x from the left), and that $f'(x+h) > 0$ if h > 0 (intuitively, f is increasing as we go right from x). Now, by the first derivative test, $f$ has a local minimum at $x$.

## 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 $\ f''(x) > 0$ and concave down if $\ f''(x) < 0$. Note that if $\ f(x) = x^4$, then $\ x=0$ has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine if a given point is an inflection point.