Question

5.2.3 quiz: polynomial functions
question 10 of 10
which of the following is most likely to be a relative maximum for this graph?
a. (3,0)
b. (-1,8)
c. (-2,0)
d. (2,-4)

Explanation:

Step1: Recall the definition of relative maximum

A relative maximum of a function is a point where the function changes from increasing to decreasing. At this point, the \(y -\)value is greater than the \(y -\)values of nearby points.

Step2: Analyze each option

• Option A: \((3,0)\)

This is an \(x -\)intercept (where \(y = 0\)). Since the function value at a relative maximum should be greater than its neighboring points' function values, and \(y = 0\) is not greater than values of nearby points (e.g., consider points to the left of \(x = 3\) where \(y>0\) in some intervals), this is not a relative maximum.

• Option B: \((-1,8)\)

Looking at the graph, around \(x=-1\), the function changes from increasing (as \(x\) approaches \(-1\) from the left) to decreasing (as \(x\) moves to the right of \(-1\)). Also, the \(y -\)value \(y = 8\) is greater than the \(y -\)values of nearby points.

• Option C: \((-2,0)\)

This is an \(x -\)intercept (\(y = 0\)). Similar to option A, \(y = 0\) is not greater than values of nearby points (e.g., consider points to the right of \(x=-2\) where \(y>0\) in some intervals), so this is not a relative maximum.

• Option D: \((2,-4)\)

At this point, the function changes from decreasing to increasing (it is a relative minimum because the \(y -\)value \(-4\) is less than the \(y -\)values of nearby points).

Answer:

B. \((-1,8)\)