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Second-Order Conditions
The second-order condition provides a sufficient condition but, as we will see, not a necessary condition, for characterizing a stationary point as a local maximum or a local minimum.
Second-Order Conditions
Local Maximum: If the second derivative of the differentiable function y=f(x) is negative when evaluated at a stationary point x* (that is, f”(x*)<0), then that stationary point represents a local maximum.
Local Minimum: If the second derivative of the differentiable function y=f(x) is positive when evaluated at a stationary point x* (that is, f”(x*)>0), then that stationary point represents a local minimum.
Second-Order Conditions for Our Examples
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Second-Order Conditions for Example 3
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Stationary Point of a strictly concave function
If the function f(x) is strictly concave on the interval (m,n) and has the stationary point x*, where m<x*<n, the x* is a local maximum in that interval. If a function is strictly concave everywhere, then it has, at most, one stationary point, and that stationary point is a global maximum.
Note that example 1 satisfies this condition.
Stationary Point of a Strictly Convex Function
If the function f(x) is strictly convex on the interval (m,n) and has the stationary point x*, where m<x*<n, the x* is a local minimum in that interval. If a function is strictly convex everywhere, then it has, at most, one stationary point, and that stationary point is a global minimum.
Note that example 2 satisfies this condition.
Inflection Point
The twice-differentiable function f(x) has an inflection point at x* if and only if the sign of the second derivative switches from negative in some interval (m,x*) to positive in some interval (x*,n), in which case the function switches from concave to convex at x*, or the sign of the second derivative switches from positive in some interval (m,x*) to negative in some interval (x*,n), in which case the function switches from convex to concave at x*. Note that, in either case, m<x<n.
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Derivatives of Our Cubic Function
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