Trouble with local extrema graph

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Homework Help Overview

The original poster seeks to sketch a graph of a continuous function on the interval [1,5] that has no local maxima or minima, despite having critical numbers at 2 and 4.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the nature of critical points and their relation to local extrema, with some suggesting that a graph can be sketched without needing a specific function. Others question how critical points can exist without corresponding local extrema.

Discussion Status

There is ongoing exploration of how to satisfy the conditions of the problem, with various suggestions on sketching the graph. Some participants have provided guidance on the characteristics of the graph, while questions about continuity and the nature of critical points remain open.

Contextual Notes

Participants are considering the implications of having critical points at specified locations while adhering to the requirement of no local extrema within the given interval.

coverticus
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Homework Statement


Sketch a graph of a function f that is continuous on [1,5] and has no local maximum and minimum, but 2 and 4 are critical numbers.


Homework Equations





The Attempt at a Solution


Knowing 2 and 4 are critical numbers, I formed the base function x[tex]^{}2[/tex]-6x+8. Not sure how to go about sketching the graph the meets the stipulations beyond this.
 
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Just draw it. No need to actually have a concrete function.
 
Also, note that x^3 has no maximum or minimum but has a critical point where? What does that point look like?

Also, are you sure it's [1,5]?
 
So just sketch a graph that has no local extrema on [1,5]? If so how are 2 and 4 critical numbers?
 
Yes it is [1,5], and x^3 has a critical point at 0, and it has a slope of zero. Correct?
 
I still need somewhat of a solid answer here, do I just sketch a graph where x=0 on [1,5] or something different? Any help here would be great.
 
You sketch a graph where x=2 and x=4 have horizontal tangents, but aren't maxes or mins.
 
does that satisfy the continuity?
 
coverticus said:
does that satisfy the continuity?

Just draw the curve y=x^3 and look what happens at x=0. Now draw a curve with two points like that.
 
  • #10
how can you do that without creating an extrema?
 
  • #11
Put your pencil on a paper at x=1. Curve up until you reach x=2 then flatten out but don't go down. Increase out of the flat part till you get to x=4, then flatten out again. Then increase some more till you get to x=5.
 

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