Approximate the critical numbers of the function shown in the graph and 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.
A) The critical number yields an absolute minimum and the critical number yields an absolute maximum.
B) The critical number yields an absolute maximum and the critical number yields an absolute minimum.
C) Both the critical numbers & yield an absolute minimum.
D) Both the critical numbers and yield an absolute maximum.
E) Both the critical numbers & yield a relative minimum.

Question

​Approximate the critical numbers of the function shown in the graph and 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. ​  ​
A) ​The critical number ​​ yields an absolute minimum and the critical number yields an absolute maximum. 
B) ​The critical number yields an absolute maximum and the critical number yields an absolute minimum. 
C) ​Both the critical numbers & yield an absolute minimum. 
D) ​Both the critical numbers and yield an absolute maximum. 
E) ​Both the critical numbers & yield a relative minimum.

Quizplus · Accepted Answer

A critical point is a point where the derivative of the function is either zero or undefined. In this case, the function has a critical point where the derivative is zero or undefined. 
From the graph, we can see that the critical points are at approximately x = -1.5 and x = 1.5. At these points, the graph changes direction, so they could be relative minima or maxima. 
However, from the given answer choices, we can see that choices A, B, D, and E all have either an absolute maximum or minimum at one or both of these critical points. This cannot be true because if there is an absolute maximum or minimum at a point, then that point must be a relative maximum or minimum as well. 
Therefore, the only valid choice is C, which states that both critical points are relative minima.

​Approximate the Critical Numbers of the Function Shown in the Graph

Approximate the Critical Numbers of the Function Shown in the Graph