MHB Is only (i) true for a continuous function f with given conditions?

[Umar]

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Hi there, I'm having trouble with the above question. Basically, I need to determine which, or all of the statements are true. I've tried coming up with different ways the function can look like to satisfy or not satisfy the statements, but have come to no luck in doing so. If anyone could assist me on this one, that would be appreciated.

 

[MarkFL]

I would begin by plotting the line $y=3$ and the 3 given points:

$(1,3),\,(4,5),\,(5,3)$

What conclusions do you now draw regarding the 3 statements?

 

[Umar]

I would begin by plotting the line $y=3$ and the 3 given points:

$(1,3),\,(4,5),\,(5,3)$

What conclusions do you now draw regarding the 3 statements?

View attachment 6011

This is a rough sketch of two possible ideas I had the function f might look like, the blue line being one possibility where f(0)>3 and f(2) > 3.

The purple line only satisfies f(2) > 3 and f(6) < 3.

 

[MarkFL]

Okay, good! (Yes)

Which, if any, of the given conditions must be true then?

 

[Umar]

Okay, good! (Yes)

Which, if any, of the given conditions must be true then?

Hmm, the only think I can think of which has to be true is the first condition... since the other ones can be changed

 

[MarkFL]

Hmm, the only think I can think of which has to be true is the first condition... since the other ones can be changed

I agree...only (i) must be true. With $f(1)=3$ and $f(4)=5$ and no other possibilities for $f(x)=3$ on $(1,4)$, then we must have $f(2)>3$. You demonstrated the other two don't have to be. :D

 

[Umar]

I agree...only (i) must be true. With $f(1)=3$ and $f(4)=5$ and no other possibilities for $f(x)=3$ on $(1,4)$, then we must have $f(2)>3$. You demonstrated the other two don't have to be. :D

Thank you so much!