get which means that if (intuitively, f is decreasing as it approaches from the left), and that if (intuitively, f is increasing as we go right from x). Now, by the first-derivative test, has a local minimum at . 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 and concave down if . Note that if , then has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is an inflection point. Higher-order derivative test 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 higher-order derivative test. Let f be a real-valued, sufficiently differentiable function on an interval , let , and let be a natural number. Also let all the derivatives of f at c be zero up to and including the n-th derivative, but with the (n + 1)th derivative being non-zero: There are four possibilities, the first two cases where c is an extremum, the second two where c is a (local) saddle point: If n is odd and , then c is a local maximum. If n is odd and , then c is a local minimum. If n is even and , then c is a strictly decreasing point of inflection. If n is even and , then c is a strictly increasing point of inflection. Since n must be either odd or even, this analytical test classifies any stationary point of f, so long as a nonzero derivative shows up eventually. Example Say we want to perform the general derivative test on the function at the point . To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero. , , , , , , As shown above, at the point , the function has all of its derivatives at 0 equal to 0, except for