Could you please explain to me the difference between the second derivative test and the procedure used to determine inflection points?
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The second derivative test is a way to determine whether a critical value (found from first derivative equals zero) is a local maximum, local minimum or point of inflection.
If dy/dx = 0 and d2y/dx2 > 0 then the point is a local minimum.
If dy/dx = 0 and d2y/dx2 < 0 then the point is a local maximum.
If dy/dx is NOT 0 and d2y/dx2 = 0 then it is a point of inflection.
The difficult case is dy/dx = 0 and d2y/dx2 = 0. This is USUALLY a horizontal point of inflection but there are rare exceptions when it can be a local maximum or local minimum, e.g. y = x^4 at the origin.
If dy/dx = 0 and d2y/dx2 > 0 then the point is a local minimum.
If dy/dx = 0 and d2y/dx2 < 0 then the point is a local maximum.
If dy/dx is NOT 0 and d2y/dx2 = 0 then it is a point of inflection.
The difficult case is dy/dx = 0 and d2y/dx2 = 0. This is USUALLY a horizontal point of inflection but there are rare exceptions when it can be a local maximum or local minimum, e.g. y = x^4 at the origin.