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greg_rack
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How do I distinguish between a point of local maxima or minima, when the second derivative in that point is equal to zero?
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Got it!snorkack said:"Stationary point" means that first derivative is zero and you specify that second also is.
In that case, if the first nonzero derivative is odd (third, fifth etc.), it is neither maximum nor minimum but a stationary inflection point. If the first nonzero derivative is even (fourth, sixth etc.), the point is minimum if that derivative is positive and maximum if negative.
The 2nd derivative being equal to 0 indicates that the slope of the graph is changing from increasing to decreasing or vice versa. This means that the point where the 2nd derivative is equal to 0 is a possible turning point, which could be a local max or min.
The 2nd derivative test involves taking the 2nd derivative of the function and evaluating it at the critical points (where the 1st derivative is equal to 0 or undefined). If the 2nd derivative is positive at a critical point, then it is a local minimum. If the 2nd derivative is negative at a critical point, then it is a local maximum.
No, the 2nd derivative test can only determine if a point is a local max/min. To determine if a point is a global max/min, you would need to evaluate the function at all critical points and the endpoints of the interval.
Yes, there are other methods such as using the 1st derivative test or graphing the function to visually identify the local max/min points. However, the 2nd derivative test is often preferred as it is more accurate and efficient.
Yes, the 2nd derivative can be equal to 0 at a point that is not a local max/min. This could happen at an inflection point where the concavity of the graph changes, but the point is not a local max/min.