Local maximum(s) and minimum(s) and point(s) of inflection right? Do I have to find any other points?
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The critical points of a function aren't necesarily local maxima or minima, but rather they are potential extrema. That is, they are all of the points on a graph in which the derivative is equal to zero or undefined. These are qualities of local extrema, but they don't guarantee that the point is that of one.
Take, for instance, the function f(x)=x^3. If you were to take a derivative of this, you would get f'(x)=3x^2. This is equal to zero at the point x=0, but this is not a local maximum or minimum. This is because, although the line tangent to that point would have a slope of zero, it doesn't change from increasing to decreasing (or vice versa). Because of this, that point qualifies as a critical point, but not a local extremum. Hope this helps.
Take, for instance, the function f(x)=x^3. If you were to take a derivative of this, you would get f'(x)=3x^2. This is equal to zero at the point x=0, but this is not a local maximum or minimum. This is because, although the line tangent to that point would have a slope of zero, it doesn't change from increasing to decreasing (or vice versa). Because of this, that point qualifies as a critical point, but not a local extremum. Hope this helps.
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Critical points are points where the derivative is zero or undefined. These can include cusps and asymptotes (though these are not strictly in the domain of the function) in addition to the points you mention. Inflection points are NOT critical points unless the tangent there is horizontal or vertical.
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That's it.
Remember first derivative for extrema and second derivative for inflection points.
Remember first derivative for extrema and second derivative for inflection points.