If f is twice differentiable, and f′(x) = 0, and f′′(x) < 0, then f attains a local maximum at x. Since a negative second derivative implies a decreasing first derivative, f′ is positive throughout (x-h,x), and negative throughout (x,x+h), for some neighborhood h. Thus f is strictly increasing from the left, and strictly decreasing to the right. It attains a local maximum at x. Similarly, f(x) is a local minimum when f′(x) = 0 and f′′(x) > 0. Since concavity determines local maximum or minimum, this is called the concavity test, or the second derivative test.
Review the example function x3-3x. We determined that the local extremes are at ±1; now we can prove the first is a maximum and the second is a minimum. The second derivative of f is 6x, which is negative when x = -1 and positive when x = 1. Of course there is no true maximum or minimum, since the cubic function is unbounded.
If both the first and second derivatives are 0, and the third is nonzero, the point is not a maximum or minimum. A nonzero third derivative implies a local maximum or minimum of the first derivative. For some neighborhood about x, the sign of the first derivative is constant. The curve increases or decreases through x. Therefore f(x) cannot be a maximum or a minimum. This is called an inflection point, as illustrated by x3 at the origin.
If the first three derivatives are 0 and the fourth is positive, we again have a local minimum. The second derivative attains a local minimum, as its first derivative is 0 and its second is positive. follow the second derivative through x. The first derivative of f is negative, then positive, and f attains a local minimum at x. A similar result holds when the fourth derivative is negative; f attains a local maximum.
By induction, If a function's first nonzero derivative is of even order, the function attains a local minimum or maximum, for a positive or negative derivative respectively. If its first nonzero derivative is of odd order, we have an inflection point, and the function is increasing or decreasing, for a positive or negative derivative respectively.