Question

indicate whether each point is a relative maximum for this graph of a polynomial function.

Explanation:

Step1: Recall the definition of a relative maximum

A relative maximum of a function is a point where the function changes from increasing to decreasing. That is, in a small interval around the point, the function value at that point is greater than the function values at neighboring points.

Step2: Analyze each point

• Point A: The function is increasing as we move from left - to - right near point A (since the graph is going up from the left - hand side towards point A). So, it is not a relative maximum.
• Point B: Just before point B (to the left of B), the function is increasing, and just after point B (to the right of B), the function is decreasing. So, in a small interval around B, \(y(B)>y(x)\) for \(x\) near B.
• Point C: The function is decreasing as we move from left - to - right near point C (since the graph is going down from the left - hand side towards point C). So, it is not a relative maximum.
• Point D: The function is increasing as we move from left - to - right near point D (since the graph is going up from the left - hand side towards point D). So, it is not a relative maximum.
• Point E: Just before point E (to the left of E), the function is increasing, and just after point E (to the right of E), the function is decreasing. So, in a small interval around E, \(y(E)>y(x)\) for \(x\) near E.
• Point F: The function is increasing as we move from left - to - right near point F (since the graph is going up from the left - hand side towards point F). So, it is not a relative maximum.
• Point G: The function is increasing as we move from left - to - right near point G (since the graph is going up from the left - hand side towards point G). So, it is not a relative maximum.

Answer:

Points \(B\) and \(E\) are relative maxima.