Second derivative test

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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[edit]

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[edit]

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[edit]

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.

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