Absolute extrema means points with maximum and minimum values for the function in the given range.

Absolute extrema may be located at local extrema and at boundaries.

To look for local extrema:

If f'(x0) exists then

Local minima at x0 if:

f'(x0) = 0 and

f''(x0) >= 0 and

f'(x) < 0 for x just before x0 and

f'(x) > 0 for x just after x0

Local maxima at x0 if:

f'(x0) = 0 and f''(x0) <= 0 and

f'(x) > 0 for x just before x0 and

f'(x) < 0 for x just after x0

Also, check all points where f'(x) does not exist for extrema.

A point of inflection at x0 is when f''(x) changes sign before and after x0

To sketch the graph, first plot the points for f(x) that are already given. Then use what you know about f'(x) to get the general increase and decrease. Use f''(x) to tell whether the slope should be increasing or decreasing.

Make sure that points with undefined derivatives are not smooth.