Concavity And The Second Derivative For Function F(X)

f”(x)=0 and changes sign

f'(x) has extrema and f(x) has an inflection point

If f”(x)=0 and changes sign, then it means that the second derivative of the function f(x) has a critical point at x, which can either be a point of inflection or a local extremum.

To determine whether the critical point is a point of inflection or a local extremum, we need to examine the behavior of the function f(x) in the neighborhood of x. This can be done by analyzing the first derivative f'(x) of the function.

If f'(x) changes sign at x, then x is a point of local extremum. If the sign change is from positive to negative, then it’s a local maximum, and if the sign change is from negative to positive, then it’s a local minimum.

If f'(x) does not change sign at x, then x is a point of inflection. A point of inflection is a point at which the second derivative changes sign, but the first derivative does not.

To summarize, if f”(x)=0 and changes sign, then x can be either a point of inflection or a local extremum. To determine which is the case, we need to examine the behavior of f'(x) in the neighborhood of x.

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