Question

multiple-choice question the graph of f(x) is shown below. which statement is correct? f(x) has a relative maximum. f(x) has a global (or absolute) minimum. f(x) does not have a relative minimum. f(x) has a global (or absolute) maximum.

Explanation:

To solve this, we analyze each option using the graph of \( f(x) \):

Step 1: Analyze "f(x) has a relative maximum"

A relative maximum is a point where the function changes from increasing to decreasing (a "peak" in the graph). From the graph, we can see a peak (where the function rises, then falls), so this statement is correct (we will confirm the others to be sure).

Step 2: Analyze "f(x) has a global (or absolute) minimum"

A global minimum is the lowest point on the entire domain of the function. The left - hand end of the graph goes downwards (towards \( -\infty \)), so there is no lowest point (the function can get arbitrarily small). Thus, this statement is incorrect.

Step 3: Analyze "f(x) does not have a relative minimum"

A relative minimum is a point where the function changes from decreasing to increasing (a "valley" in the graph). The graph has a valley (where the function falls, then rises), so there is a relative minimum. Thus, this statement is incorrect.

Step 4: Analyze "f(x) has a global (or absolute) maximum"

A global maximum is the highest point on the entire domain of the function. The right - hand end of the graph goes upwards (towards \( +\infty \)), so there is no highest point (the function can get arbitrarily large). Thus, this statement is incorrect.

Answer:

The correct statement is: \( \boldsymbol{f(x) \text{ has a relative maximum}} \) (the option corresponding to this statement).