Question

Approximate the critical numbers of the function shown in the graph. Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown. (Enter your answers as a comma-separated list.)

Answer 1

A relative maximum point is a point where the function changes direction from increasing to decreasing (making that point a "peak" in the graph). Similarly, a relative minimum point is a point where the function changes direction from decreasing to increasing (making that point a "bottom" in the graph).

An absolute maximum point is a point where the function obtains its greatest possible value. Similarly, an absolute minimum point is a point where the function obtains its least possible value.

Our function starts increasing, and approximately at x = 2, it starts decreasing, which means this is a relative maximum point, and this is also the greatest possible value for our function, therefore, this is also an absolute maximum. There are no further critical points.