Relative maximum on a horizontal line

[Alex L.]

If I have a line that crosses the x-axis and is horizontal, would it be considered a relative maximum? It's a short and sweet question, not sure what else to add. Thank you for your time!

 

[LCKurtz]

Since the graph is given to be represented by a not identically zero polynomial, there can be no interval of positive length where the graph is horizontal.

 

[lev888]

View attachment 17921If I have a line that crosses the x-axis and is horizontal, would it be considered a relative maximum? It's a short and sweet question, not sure what else to add. Thank you for your time!

Define relative maximum.

 

[Dr.Peterson]

View attachment 17921If I have a line that crosses the x-axis and is horizontal, would it be considered a relative maximum? It's a short and sweet question, not sure what else to add. Thank you for your time!

At each indicated point, the tangent to the curve is horizontal, and that point is higher than any other point on the graph. That is all that matters in the question. These points are both relative maxima because they are both higher than any point nearby; but the question is about the absolute maximum. They are both that, too.

I wouldn't say that this "line" (by which I think you mean "curve") "crosses" the x-axis, but that it intersects (and turns back). That's just a matter of English. The important thing in the question is that they are asking not about the location of the maximum, but its value. The maximum value is the value of y at that point, and it is the same for both maximum points. So there is one maximum value, namely the number 0.

 

[Otis]

Typo in exercise: "values" is supposed to be "value(s)".