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The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
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 ...
If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold. For example, the second derivative of f ( x ) = x 4 {\displaystyle f(x)=x^{4}} is f ″ ( x ) = 12 x 2 {\displaystyle f''(x)=12x^{2}} , which is zero for x = 0 , {\displaystyle x=0,} but x 4 {\displaystyle x^{4}} is strictly ...
A cubic function is concave (left half) when its first derivative (red) is monotonically decreasing i.e. its second derivative (orange) is negative, and convex (right half) when its first derivative is monotonically increasing i.e. its second derivative is positive
At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)
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
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Its first derivative ranges monotonically in the open interval from the infimum to the supremum of the support of the probability distribution, and its second derivative is strictly positive everywhere it is defined, except for the degenerate distribution of a single point mass.