Question

The continuous function h(x) has exactly one critical point. Find the x-values at which the global maximum and the global minimum occur in the interval 1≤x≤4.

h′(2) is undefined, h′(x)=-1 for x<2 and h′(x)=1 for x>2.


Enter the exact answers.


The global maximum occurs when x=
.


The global minimum occurs when x=

Answer 1

Answer: Since h(x) has only one critical point, it must occur at x = 2. Therefore, we need to examine the behavior of h(x) to the left and right of x = 2 to determine the global maximum and minimum.

To the left of x = 2, h′(x) = -1, which means that h(x) is decreasing. Therefore, the global maximum must occur at x = 2.

To the right of x = 2, h′(x) = 1, which means that h(x) is increasing. Therefore, the global minimum must occur at x = 4.

Therefore, the exact answers are:

The global maximum occurs when x= 2.

The global minimum occurs when x= 4.