Question

the given function f(x) is defined on the interval -9, 9) and represented by the graph below. for the circled point, select all that apply.

answer attempt 1 out of 2
the point is a global maximum
the point is a global minimum
the point is a local maximum
the point is a local minimum
the point is neither a global nor a local extremum

Explanation:

1. Global Maximum: A global maximum is the highest point on the entire domain. The left - most point (the black dot on the left) has a higher y - value than the circled point, so the circled point is not a global maximum.
2. Global Minimum: A global minimum is the lowest point on the entire domain. The lowest point on the graph (the black dot at the bottom of the right - hand curve) has a lower y - value than the circled point, so the circled point is not a global minimum.
3. Local Maximum: A local maximum is a point where the function value is greater than the values of the function in its immediate neighborhood. Looking at the circled point, to the left of it, the function is increasing towards it, and to the right of it, the function is decreasing (since the graph goes down after the circled point). So, in its local neighborhood, it is the highest point, so it is a local maximum.
4. Local Minimum: A local minimum is a point where the function value is less than the values of the function in its immediate neighborhood. Since the function is increasing before the circled point and decreasing after, it is not a local minimum.
5. Neither: Since it is a local maximum, it is not neither.

Answer:

The point is a local maximum