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The second-derivative test for functions of one and two variables is simpler than the general case. In one variable, the Hessian contains exactly one second derivative; if it is positive, then x {\displaystyle x} is a local minimum, and if it is negative, then x {\displaystyle x} is a local maximum; if it is zero, then the test is inconclusive.
The higher-order derivative test or general derivative test is able to determine whether a function's critical points are maxima, minima, or points of inflection for a wider variety of functions than the second-order derivative test. As shown below, the second-derivative test is mathematically identical to the special case of n = 1 in the ...
Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.
I. Nonstiff Problems. Springer Series in Computational Mathematics. Vol. 8 (2nd ed.). Springer-Verlag, Berlin. ISBN 3-540-56670-8. MR 1227985. Ernst Hairer and Gerhard Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, second edition, Springer Verlag, Berlin, 1996. ISBN 3-540-60452-9.
First derivative test; Second derivative test; Extreme value theorem; Differential equation; Differential operator; Newton's method; Taylor's theorem; L'Hôpital's rule; General Leibniz rule; Mean value theorem; Logarithmic derivative; Differential (calculus) Related rates; Regiomontanus' angle maximization problem; Rolle's theorem
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum.
Let = be an infinite series with real terms and let : be any real function such that (/) = for all positive integers n and the second derivative ″ exists at =. Then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} converges absolutely if f ( 0 ) = f ′ ( 0 ) = 0 {\displaystyle f(0)=f'(0)=0} and diverges otherwise.