Local maximum - the curve y=f(x) will have a local maximum at a point if the tangent to the graph has zero slope and the graph changes from its increasing nature to decreasing.
In this question, we can find if the local maximum exists at that point by finding the nature of the second derivative of the function. If it is positive, then there exists a local maximum at that pt.
Local minimum - the curve y=f(x) will have a local minimum at a point if the tangent to the graph has zero slope and the graph changes from its decreasing nature to increasing.
In this question, we can find if the local minimum exists at that point by finding the nature of the second derivative of the function. If it is negative, then there exists a local minimum at that pt.
Horizontal point of inflection - If the second derivative of the function at that pt. is zero then that pt. is the pt. of inflection.
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