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For a function f, if its second derivative f″(x) exists at x 0 and x 0 is an inflection point for f, then f″(x 0) = 0, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third ...
If ″ =, the second derivative test says nothing about the point , a possible inflection point. The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration.
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function.
If the second derivative is null, the critical point is generally an inflection point, but may also be an undulation point, which may be a local minimum or a local maximum. For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point.
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 is a local minimum, and if it is negative, then is a local
Visually, a twice differentiable convex function "curves up", without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold.
A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope. [3] [4] Points where concavity changes (between concave and convex) are inflection points. [5]
Although the first derivative (3x 2) is 0 at x = 0, this is an inflection point. (2nd derivative is 0 at that point.) Unique global maximum at x = e. (See figure at right) x −x: Unique global maximum over the positive real numbers at x = 1/e. x 3 /3 − x: First derivative x 2 − 1 and second derivative 2x.