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second derivative

Second derivative test

In calculus, a branch of mathematics, 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.

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

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

In the last case, the function may have a local maximum or minimum there, but the function is sufficiently "flat" that this is undetected by the second derivative. Such an example is f(x) = x ^4.

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 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 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, and
f'(x+h) > 0 if h > 0.

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.

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References

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