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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

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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, $f^{\prime }(x)=0$. Then

$0<f^{″}(x)=\underset{h\to 0}{lim}\frac{f^{\prime }(x+h)-f^{\prime }(x)}{h}=\underset{h\to 0}{lim}\frac{f^{\prime }(x+h)-0}{h}=\underset{h\to 0}{lim}\frac{f^{\prime }(x+h)}{h}.$

Thus, for h sufficiently small we get

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

which means that $f^{\prime }(x+h)<0$ if h < 0 (intuitively, f is decreasing as it approaches x from the left), and that $f^{\prime }(x+h)>0$ 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.

See also

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

References

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