Problem : On the graph below, find the local and absolute extrema on the interval [a, b]

The critical points occur where the tangent is horizontal, at points c, d, and e listed below. c and e are local minima. d is a local maximum. The endpoint b is the absolute maximum, and the critical point e is the absolute minimum in this interval.

Problem : Find the critical points of f (x) = x3 + x2

f'(x) = x2 + 2x
f'(x) = 0 at x = 0 and x = - 2

Problem : Does f (x) = x3 + x2 have an absolute maximum?

No, it does not, since as x approaches infinity, f (x) approaches infinity, so the function grows without bound and has no maximum.

Problem : The Extreme Value Theorem doesn't apply to continuous functions on open intervals, but could such a function have both an absolute minimum and an absolute maximum on that open interval?

Yes, it certainly could. These extrema could occur at critical point on the interior of the interval:

Problem : Do absolute extrema always count as local extrema?

No. Absolute extrema that occur at the endpoints of an interval are not considered local extrema because the definition of a local extremum requires that there be an open interval I containing the extremum, but if the point is an endpoint, no such open interval exists.