To sketch f between a and b.

Step 1 Note the location of any singular points of f, and the behavior of f at a,b and near the singular points.

Step 2 Calculate f ' and f ''.

Step 3 Note where f ' is < 0, = 0, > 0 These correspond to regions in which f is decreasing, stationary, increasing respectively.

A critical point is one where f '(x) = 0.

Step 4 Note where f '' is < 0, > 0 

These correspond to regions in which f is concave down and concave up respectively (if you forget which corresponds to which, refer to the function f(x) = x² ). An inflection point is one at which f ''(x) = 0 and changes sign across x.