Trouble with local extrema graph

[coverticus]

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$$^{}2$$-6x+8. Not sure how to go about sketching the graph the meets the stipulations beyond this.

 

[JasonRox]

Just draw it. No need to actually have a concrete function.

 

[JasonRox]

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]?

 

[coverticus]

So just sketch a graph that has no local extrema on [1,5]? If so how are 2 and 4 critical numbers?

 

[coverticus]

Yes it is [1,5], and x^3 has a critical point at 0, and it has a slope of zero. Correct?

 

[coverticus]

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.

 

[Dick]

You sketch a graph where x=2 and x=4 have horizontal tangents, but aren't maxes or mins.

 

[coverticus]

does that satisfy the continuity?

 

[Dick]

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.

 

[coverticus]

how can you do that without creating an extrema?

 

[Dick]

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.

 

[coverticus]

ok thanks, if you feel so inclined you could help me with another question I have on here. https://www.physicsforums.com/showthread.php?t=194855