The Significance Of A Negative Second Derivative For Concavity And Maxima Of Functions

f”(x)=0 and changes sign

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

If f”(x) = 0 and changes sign, it means that the second derivative of the function has a critical point at x and changes from positive to negative or from negative to positive. This critical point can either be a point of inflection or a maximum/minimum point.

To determine whether this critical point is a maximum/minimum or a point of inflection, you can look at the sign of the first derivative f'(x) on either side of the critical point. If f'(x) changes sign at the critical point, then it is a maximum/minimum. If f'(x) does not change sign at the critical point, then it is a point of inflection.

If f”(x) changes sign from positive to negative, then it means that the function is concave up before the critical point and concave down after the critical point. This indicates a maximum point. Conversely, if f”(x) changes sign from negative to positive, then it means that the function is concave down before the critical point and concave up after the critical point. This indicates a minimum point.

In summary, if f”(x) = 0 and changes sign:
– Determine whether the critical point is a maximum/minimum or a point of inflection by looking at the sign of f'(x).
– If f”(x) changes sign from positive to negative, it indicates a maximum point.
– If f”(x) changes sign from negative to positive, it indicates a minimum point.

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