Search results
Results From The WOW.Com Content Network
If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point . Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second ...
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 f ″ ( x ) > 0 {\displaystyle f''(x)>0} and concave down if f ″ ( x ) < 0 ...
The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave ...
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 ...
The function has second derivative ; thus it is convex on the set where and concave on the set where Examples of functions that are monotonically increasing but not convex include f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} and g ( x ) = log x {\displaystyle g(x)=\log x} .
The specific nature of a stationary point at x can in some cases be determined by examining the second derivative f″(x): If f″(x) < 0, the stationary point at x is concave down; a maximal extremum. If f″(x) > 0, the stationary point at x is concave up; a minimal extremum.
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.
A concave function is also synonymously called concave downwards, concave down, convex upwards, ... Similarly, the derivative of the second derivative, ...