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| In [[calculus]], the '''second derivative test''' is a criterion for determining whether a given [[critical point (mathematics)|critical point]] of a real function of one variable is a local [[Maxima and minima|maximum]] or a local [[Maxima and minima|minimum]] using the value of the [[second derivative]] at the point.
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| The test states: if the function ''f'' is twice [[differentiable]] at a critical point ''x'' (i.e. ''f'(x) = 0''), then:
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| * If <math>\ f^{\prime\prime}(x) < 0</math> then <math>\ f</math> has a local maximum at <math>\ x</math>.
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| * If <math>\ f^{\prime\prime}(x) > 0</math> then <math>\ f</math> has a local minimum at <math>\ x</math>.
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| * If <math>\ f^{\prime\prime}(x) = 0</math>, the test is inconclusive.
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| In the latter case, [[Taylor's theorem#Taylor's theorem in one real variable|Taylor's Theorem]] may be used to determine the behavior of ''f'' near ''x'' using [[higher derivative]]s.
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| ==Multivariable case==
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| {{main|Second partial derivative test}}
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| For a function of more than one variable, the second derivative test generalizes to a test based on the [[eigenvalue]]s 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 matrix|singular]], then the second derivative test is inconclusive.
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| ==Proof of the second derivative test==
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| Suppose we have <math>f''(x) > 0</math> (the proof for <math>f''(x) < 0</math> is analogous). By assumption, <math>f'(x) = 0</math>. Then
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| :<math>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}.</math>
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| Thus, for ''h'' sufficiently small we get
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| :<math>\frac{f'(x+h)}{h} > 0</math>
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| which means that
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| <math>f'(x+h) < 0</math> if ''h'' < 0 (intuitively, f is decreasing as it approaches ''x'' from the left), and that <math>f'(x+h) > 0</math> if ''h'' > 0 (intuitively, f is increasing as we go right from ''x''). Now, by the [[first derivative test]], <math>f</math> has a local minimum at <math>x</math>.
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| == Concavity test ==
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| A related but distinct use of second derivatives is to determine whether a function is [[Concave function|concave up]] or [[Concave function|concave down]] at a point. It does not, however, provide information about [[inflection points]]. Specifically, a twice-differentiable function ''f'' is concave up if <math>\ f''(x) > 0</math> and concave down if <math>\ f''(x) < 0</math>. Note that if <math>\ f(x) = x^4</math>, then <math>\ x=0</math> 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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| == See also ==
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| *[[Hessian matrix#Bordered Hessian|Bordered Hessian]]
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| * [[First derivative test]]
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| * [[Optimization (mathematics)]]
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| * [[Fermat's theorem (critical points)|Fermat's theorem]]
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| * [[Higher-order derivative test]]
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| * [[Differentiability]]
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| * [[Extreme value]]
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| * [[Convex function]]
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| ==References==
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| * [http://mathworld.wolfram.com/SecondDerivativeTest.html "Second Derivative Test" at Mathworld]
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| * [http://www.math.hmc.edu/calculus/tutorials/secondderiv/ Concavity and the Second Derivative Test]
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| *[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=606&bodyId=948 Thomas Simpson's use of Second Derivative Test to Find Maxima and Minima] at [http://mathdl.maa.org/convergence/1/ Convergence]
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| [[Category:Differential calculus]]
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