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Now, the second derivative of the function tells us the rate of change of the first derivative. It therefore follows that if the second derivative is positive at the stationary point, then the gradient is increasing. The fact that it is a stationary point in the first place means that this can only be a minimum. Conversely, if the second derivative is negative at that point, then it is a maximum. Now, if the second derivative is zero, we have a problem. It could be a point of inflexion, or it could still be an extremum. Examples of each of these cases are below-all have a second derivative equal to zero at the stationary point in question: 1  has a point of inflexion at.
As discussed at the end of Lesson 13.1, x = 0 is a critical point of f (x) = x 3 but there is no local extremum there. It was noted that the first derivative (slope of the tangents) is positive on both sides of the critical point, so the sign of the derivative does not change from one side of x = 0 to the other.art III If the first derivative has the same sign on both sides of the critical point, then the function does not have a local maximum or a local minimum at the critical point. Extrema at Endpoints A function's first derivative can only be evaluated on one side of an endpoint of a function's domain.