Question #66306

why is the interpretation of the second order derivative opposite to their sighns

Expert's answer

One definition of an inflection point is where the second derivative changes sign (from positive to negative or the reverse).

The Intermediate Value Theorem for derivatives says that if a derivative is defined on a closed interval then it assumes any intermediate value between the values at the endpoints of that interval. The second derivative is obviously a derivative, so if it is defined on any interval including the inflection point, if we look at a value on one side of the inflection point, which must be positive, and another value on the other side, which must be negative, there is some place where the second derivative is zero. The only place it can be zero is at the inflection point. Therefore, it is commonly said that the second derivative at the inflection point must be zero.

Reference:

http://math.stackexchange.com/questions/1334248/why-is-the-second-derivative-of-an-inflection-point-zero

The Intermediate Value Theorem for derivatives says that if a derivative is defined on a closed interval then it assumes any intermediate value between the values at the endpoints of that interval. The second derivative is obviously a derivative, so if it is defined on any interval including the inflection point, if we look at a value on one side of the inflection point, which must be positive, and another value on the other side, which must be negative, there is some place where the second derivative is zero. The only place it can be zero is at the inflection point. Therefore, it is commonly said that the second derivative at the inflection point must be zero.

Reference:

http://math.stackexchange.com/questions/1334248/why-is-the-second-derivative-of-an-inflection-point-zero

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