f”(x)=0 and changes sign
f'(x) has extrema and f(x) has an inflection point
If the second derivative f”(x) is equal to zero and changes sign:
1. The point x is a point of inflection.
2. The concavity changes from concave up to concave down or vice versa at x.
The first derivative, f'(x), will give the slope of the tangent line to the curve at x. If f”(x) is positive, the curve is concave up, and if f”(x) is negative, the curve is concave down. A zero value of f”(x) indicates that the curve is changing concavity.
Therefore, if f”(x) changes sign and is equal to zero, the second derivative test may be inconclusive. However, we know that x is a point of inflection since the concavity changes direction. At a point of inflection, the tangent slope changes from increasing to decreasing or vice versa. This means that the curve is not changing direction but rather changing the rate of change at x.
In summary, if f”(x) is equal to zero and changes sign, we can conclude that the point x is a point of inflection on the curve, and the concavity of the curve changes at x from concave up to down or vice versa.
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